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Matter, energy, force, and work; a plain 
 
 3 1924 012 401 695 
 
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MATTER, ENERGY, FORCE, AND WORK 
 
MATTER, ENERGY, FORCE 
 AND WORK 
 
 A PLAIN PRESENTATION OF FUNDAMENTAL 
 
 PHYSICAL CONCEPTS AND OF THE 
 
 VORTEX-ATOM AND OTHER 
 
 THEORIES 
 
 SILAS W. HOLMAN 
 
 PROFESSOR OF PHYSICS (EMERITUS) MASSACHUSETTS 
 INSTITUTE OF TECHNOLOGY 
 
 TStia g0i* 
 THE MACMILLAN COMPANY 
 
 LONDON : MACMILLAN & CO., Ltd. 
 1898 
 
 All rights reaer'oed 
 
COPYEIQBT, 1898, 
 
 By SILAS WHITCOMB HOLMAM". 
 
 Wortooolr 33reBa 
 
 J. S. CuahinK & Co. — Berwick & Smith 
 Norwood Mass. U.S.A. 
 
S$R&mdiu6ttt6 ^n&titutt of tKecljnolog?, 
 
 TO THE MEMBERS, PAST AND PRESENT, OP ITS CORPORATION AND 
 
 FACULTY, AND TO ITS ALUMNI, 
 
 AND BECAUSE OF WHAT, INDIVIDUALLY AND COLLECTIVELT, 
 
 THEY HAVE BROUGHT INTO MY LIFE, 
 
 1 IBrttcatE 
 
 WHATEVER OF GOOD THIS VOLUME MAY CONTAIN. 
 
PREFACE. 
 
 The aim of this book, prepared during a period of en- 
 forced quiet, is to present in a plain and logical manner 
 some fundamental ideas and definitions of physics. The 
 purpose is not to set forth the experimental side of the 
 subject, nor to describe phenomena or laws ; the intention 
 is rather to assume a slight knowledge of these, and to 
 proceed in an orderly manner to develop the concepts and 
 definitions. 
 
 Engineers and members of the other technical professions 
 will find here, it is hoped, an aid to clearer thinking in 
 their practical dealing with the subjects treated. Now, as 
 never before, correct views of energy, force, and work are 
 essential in these professions, through the progress of the 
 application of science to the industrial arts. The extent to 
 which these applications are daily brought home to the un- 
 technical members of the community is likewise so great 
 that acquaintance with their underlying ideas is no less a 
 matter of self-interest than of education. The book natu- 
 rally addresses itself most directly, however, to students 
 and teachers in physics and chemistry, whether elementary 
 or advanced, and of the natural sciences in general. 
 
 To have its greatest educational value, physics must be 
 presented in logical sequence. The fundamental definitions 
 must follow each other in such a manner that each involves 
 but one important idea or proposition in advance of the 
 preceding one. Every term must be clearly defined before 
 being used, and no familiar word should be employed in a 
 
viii PREFACE. 
 
 technical sense without previous explanation. Only in this 
 way will the student be assured that at every stage of his 
 progress he possesses a secure and stable body of knowledge, 
 and is not working in a circle. Benefit will be received 
 from the logical habit of thought thus engendered, not less 
 than from the knowledge acquired. 
 
 The first of the two parts into which the work has been 
 divided contains its subject-matter proper. The definitions 
 and views given are not designed to be an exposition of 
 current and accepted usage or literature. Rather they con- 
 stitute a sporadic attempt at clear, consecutive setting forth 
 of individual thought. The book is constructive in spirit, 
 not critical ; and it is purposely almost devoid of historical 
 and personal allusions. 
 
 Part second consists of summaries of the chief theories of 
 the nature of matter, energy, and force. These are intended 
 mainly to give more concreteness to the concepts than 
 could properly be introduced into the first part. Their 
 purely hypothetical character demands this clear demarca- 
 tion, but by no means debars them from service in the 
 present connection. 
 
 The publication of the volume has been rendered possible 
 by the generous kindness of Mr. Francis Blake, of Boston. 
 The possibility of its preparation, as well as of all other 
 work during several years, has been due to the personal 
 devotion, care, and counsel of Mrs. Holman, and to her 
 cooperation, which has included, as a part, the reading to 
 me of all references consulted and of all the manuscript 
 and proofs for revision. Full acknowledgment is tendered 
 to Lord Kelvin and to Professor J. J. Thomson for per- 
 mitting the use of extracts from personal letters ; to Miss 
 Catharine B. Eunkle for friendly aid and criticism ; and to 
 my friends Professors H. M. Goodwin and A. A. Noyes for 
 criticism and suggestion. 
 
 SILAS AV. HOLMAN. 
 
 Boston, August, 1898. 
 
CONTENTS. 
 
 PART I. 
 
 CHAPTER I. 
 
 Substance, Matter. 
 
 PAGE 
 
 Substance, Body, Object 3 
 
 Properties . . 6 
 
 Substances, Elements .7 
 
 Matter 10 
 
 Mass 11 
 
 Weightal 11 
 
 CHAPTER II. 
 
 Motion. 
 
 Definitions 12 
 
 CHAPTER III. 
 Energy. 
 
 Laws of Motion 18 
 
 Inertia 19 
 
 Definition of Energy 19 
 
 Muscular Energy . . 22 
 
 CHAPTER IV. 
 
 EoEMs OF Energy. 
 
 Recognition of Forms of Energy . 25 
 
 Kinetic Energy 25 
 
 Kinetic Energy of Rotation 27 
 
 ix 
 
coy TENTS. 
 
 Molar, Molecular, Atomic Kinetic Energy . . . 27 
 
 Gravitation Energy ... 28 
 
 Heat .... . . .29 
 
 Energy of Elasticity . . . 30 
 
 Cohesion Energy . . 50 
 
 Chemical Energy . . 31 
 
 Electrical Energy ... . .33 
 
 Magnetic Energy ... .34 
 
 Energy of Sound Vibrations . . . . 3i 
 
 Itadiant Energy : Light, Radiant Heat, Electro-magnetic Radi- 
 ation ... . . . . . .'J.j 
 
 Muscular Energy . . 36 
 
 List of Forms of Energy ... 37 
 
 Classification of Energy . . ... 38 
 
 Mechanical Energy . . . . 38 
 
 CHAPTER V. 
 
 FOKCE. 
 
 Definition of Force . . . 40 
 
 How Energy produces Force . 43 
 
 Rooognition of Forces . . 44 
 
 Resistance . . . 45 
 
 External Friction of Solids . . 46 
 
 Internal Friction of Solids . 47 
 
 Internal Friction of Fluids . 49 
 
 Stress, Attraction, Repulsion 50 
 
 Force as an Index of Energy . 52 
 
 Remarks upon Force .... ... 52 
 
 CHAPTER VI. 
 
 Kinetic Energy, Kinekgety. 
 
 Apparatus . 55 
 
 Quantity of Kinetic Energy ... 58 
 
 Capacity for Kinetic Energy : Kinergety . 59 
 
 Kinergety vs. Mass ... 61 
 
 Equal Kinergety ... .... 62 
 
 Graded Set of Kinergeties or Masses 62 
 
CONTENTS. xi 
 
 PAGE 
 
 Measurement of Kinergety 62 
 
 Law of Kinetic Energy 64 
 
 Primary iVteasurement of Kinetic Energy 65 
 
 CHAPTER VII. 
 
 Enekgy (Continued). 
 
 Transformation of Energy ... ... 67 
 
 Conservation of Energy ... .67 
 
 Criterion for Energy . . .... .72 
 
 Historical Slietcli .... .... 73 
 
 Otlier Statements of the Principle . . . 77 
 
 Examples of tiie Transformation of Energy . 78 
 
 Gravitation and Kinetic Energy . . 78 
 
 Gravitation and Heat . . . .83 
 
 Kinetic Energy into Heat . . ,83 
 
 Electrical Energy into Kinetic Energy . 84 
 
 Electrical Energy into Heat . . 85 
 
 Chemical Energy into Heat .... . 85 
 
 Chemical Energy into Electrical Energy . . 87 
 
 Magnetic Energy into Kinetic Energy ... 88 
 
 Heat into Mechanical Energy . . . .88 
 
 Electrical Energy into Mechanical Energy . . 89 
 
 Remarks on the Transformation of Energy . . 90 
 
 Further Remarks on the Definition of Energy . 92 
 
 Exertion of Force by Energy ... 92 
 
 Dissipation of Energy 94 
 
 CHAPTER VIII. 
 
 Force Measurement. 
 
 Quantity of Force ... .96 
 
 Practical Measurement of Forces .... .98 
 
 Spring Balance ... 99 
 
 Standards of Force . . . 100 
 
 Unit of Force ... . . .105 
 
 "Weight proportional to Kinergety . . 109 
 
 Weightal . .111 
 
Xll 
 
 CONTENTS. 
 
 CHAPTER IX. 
 
 WOKK. 
 
 PAGE 
 
 Nature of Work 112 
 
 Work by and upon 118 
 
 Quantity of Work 118 
 
 Units of Work 119 
 
 Equilibrium of Energy 120 
 
 Examples of Work . 120 
 
 Power .... 122 
 
 Strength ... . .... 123 
 
 Muscular Work .123 
 
 Methods and Units of Energy Jleasurement .... 124 
 
 CHAPTER X. 
 
 Potential Enekgt. 
 
 Definition of Potential Energy 126 
 
 Examples of Potential Energy 128 
 
 Quantity of Potential Energy 131 
 
 CHAPTER XI 
 
 Matter. 
 
 Distinction between Matter and Substance 
 
 Definition of Matter 
 
 Matter Non-resistant .... 
 
 Properties of Matter . 
 
 Mass : Quantity of Matter 
 
 Practical Measurement of Mass 
 
 Weightal vs. Weight vs. Mass 
 
 Molecules and Atoms . 
 
 Conservation of Matter 
 
 135 
 140 
 140 
 142 
 147 
 151 
 152 
 155 
 156 
 
CONTENTS. Xlll 
 
 CHAPTER XII. 
 Comments ox Certain Definitions. 
 
 PAGE 
 
 Remarks ... . ... 158 
 
 On Definitions of Matter . . . 159 
 
 On Definitions of Energy . . . . . 160 
 
 On Definitions of Force 163 
 
 PART II. 
 
 SPECULATIONS ON MATTER AND ENERGY. 
 
 CHAPTER I. 
 
 Function op Theokt and Hypothesis. 
 
 Value of Theory . . . 160 
 Purpose of this Section . 170 
 
 CHAPTER II. 
 
 Kinetic Theory of Gases. 
 
 Statement of the Theory 172 
 
 Comment 181 
 
 CHAPTER III. 
 Le Sage's Theory of Gravitation. 
 
 Statement . .... . . 182 
 
 Objections . . 186 
 
 CHAPTER IV. 
 
 The Vortex-Atom Theory. 
 
 Vortex Motion .... .... 193 
 
 Application to the Explanation of Matter .... 201 
 
 Application to Gases 202 
 
XIV 
 
 CONTENTS. 
 
 Applications in Cliemistry 
 
 Dissociation . 
 
 Gravitation 
 
 Coliesion 
 
 Elasticity 
 
 Kemarks 
 
 PAG15 
 
 205 
 214 
 215 
 217 
 218 
 21!) 
 
 CHAPTER V. 
 
 Nature of Energv and Matter. 
 
 Is all Energy Kinetic 
 
 On Chemical Energy . 
 
 On the Ether 
 
 On the Nature of Force 
 
 On the Nature of Matter and Kinergety 
 
 227 
 235 
 246 
 248 
 249 
 
 Alphabetical Index . 
 
 253 
 
Part I. 
 
 FUNDAMENTAL PHYSICAL CONCEPTS. 
 
" What is mind ? ISTo matter. 
 What is matter ? Never mind." 
 
 Attributed to T. H. Key. 
 
CHAPTER I. 
 SUBSTANCE, MATTER. 
 
 Substance, Body, Object. — It may be accepted as a start- 
 ing-point in physical inquiry that our knowledge of things 
 external to ourselves, as well as of our own bodies, is based 
 upon impressions produced upon the brain by the several 
 senses. These impressions are interpreted by the aid of 
 Eeason, thus becoming ideas, or giving rise to ideas and 
 knowledge. 
 
 We thus observe that different portions of the space about 
 us affect our senses differently, and that these diverse char- 
 acteristics are more or less permanently associated with the 
 same portions of space. The apparent behavior of the 
 lower animals shows that they are similarly affected. 
 
 Further, we observe that various portions of space possess 
 the power not only to affect our senses, but also to affect 
 one another (using power in its primary sense of ability to 
 do) ; that is, we discover changes taking place simultar 
 neously in two or more portions of space, changes which 
 ■appear to stand to one another in the relation which we 
 commonly denote by the terms cmise and effect. 
 
 The direct result of these primary observations is, then, 
 that the space about us may be marked off into definitely 
 bounded portions, each of which is characterized by its own 
 group of powers. The operation and nature of these powers 
 form the chief part of the study of physics. These powers 
 are also called properties. Also, in general, any one of 
 these groups may be made to change its location relatively 
 to the others; that is, may be moved, the size and shape 
 
 3 
 
4 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 of the space occupied by the group sometimes undergoing 
 change during tlie changing of position. 
 
 To indicate what is meant by the " powers " or " prop- 
 erties " referred to, a few of them may be specifically men- 
 tioned. It is observed that any portion of space which can 
 be in any way distinctly recognized, possesses the power, 
 when in motion relatively to another recognizable portion, 
 of imparting motion to that other portion, always sacrificing 
 some of its own speed in so doing. It is found that any 
 portion appears to possess the power of approaching that 
 other portion called the earth ; that is, that it will approach 
 unless obviously prevented by the intervention of the power 
 of some other portion. This power we know familiarly as 
 gravitation or weight. Many portions have the power of 
 resisting a pull tending to separate them : cohesion. Some 
 portions not only resist any attempt to change their form, 
 but also will return to their original form when again left to 
 themselves : a power called elasticity. Certain portions 
 when brought into contact, are found to undergo sponta- 
 neously an entire change of general appearance and of many 
 of their powers : this is known as chemical action and is 
 ascribed to a corresponding power vaguely termed chemical 
 affinity. Without further description, these powers may be 
 enumerated. They are nine in number ; viz., power due to 
 motion ; power of gravitation ; heat power ; elastic power ; 
 cohesive (and adhesive) power ; chemical power ; electrical 
 power ; magnetic power ; and finally power in the form in 
 which electromagnetic action, " light," and " radiant heat " 
 are propagated through space. This list includes all the 
 " powers " properly so called, but they are probably not all 
 distinct from each other. The classification is merely a 
 provisional one, based on the customary grouping of physical 
 phenomena. There are other so-called powers, but they are 
 either different phases of the foregoing, or are not active 
 powers but passive characteristics which might better be 
 
Chap. I.] SUBSTANCE, MATTEB. 5 
 
 called by another nanie. It might perhaps be more logical 
 at this stage of the presentation of the subject, to remain 
 content with regarding these groups as mere collocations of 
 powers or properties. This abstract concept. is however 
 quite repugnant to the habit of thought engendered by the 
 entire training of early life, and is not in accord with the 
 traditions of physics. Instead of this we are accustomed 
 to regard these groups of powers as constituting attributes 
 of something which occupies, not necessarily wholly filling, 
 the particular portions of space considered, and which has 
 a more or less permanent existence outside of and inde- 
 pendent of our minds. We therefore proceed to infer that 
 there is in space something which is endued with these 
 powers. To this we give the name of substance. We 
 might then define substance as that which is inferred as 
 existing in space, and as endued with powers to affect the 
 human senses and portions of itself. Assuming however 
 that we may exclude from purely physical definitions any 
 allusion to sensation, since we have recognized at the outset 
 that it is only through sensation that we are aware of 
 physical action, this formula may be reduced to: — 
 
 Substance is that which is infeeeed as existing in 
 Space, and as endued with Powees to affect Poetions 
 OF Itself. 
 
 This definition makes no assumption as to the possibility 
 or impossibility of the entire removal of all of any one 
 power from any portion of substance, nor as to the possi- 
 bility of the existence of any of the powers apart from 
 substance. ISTeither does the definition afBrm or deny that 
 substance fills any portion of space. Whether it does so is 
 a topic for later inquiry ; and whether existence without 
 the occupancy of space is conceivable, is a metaphysical 
 question with which we are not now concerned. 
 
 Indeed it may well be remarked here that psychological 
 
6 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 and philosophical considerations as to the interpretation to 
 be placed upon such terms as existence and reality form no 
 part of physics. The terms of this science, if consistent 
 ■with one another, express a definite body of knowledge. 
 Whatever interpretation, from that of Berkeley to that of 
 the materialist, be put upon this knowledge, this interpreta- 
 tion involves ideas outside of the domain of physics, and 
 the same is true of the other natural sciences. The student 
 of these sciences cannot bear too clearly in mind this fact. 
 Always looking forward to an ultimate adequate interpretar 
 tion of the relation of the mental to the physical, he may 
 rest in the assurance that this interpretation will not vitiate 
 any real physical science, but will only illuminate it. 
 
 A Body is ant Designated Portion of Substance. 
 
 The boundary of a body may be and very often is wholly 
 imaginary. The size may be anything whatever, so long as 
 the characteristics of substance remain. When the boundary 
 is natural and not imaginary, the body may be called an 
 object. 
 
 Properties. — By quantitative examination it is easily dis- 
 coverable that some or all of the powers of a body undergo 
 change in amount from time to time. These changes ac- 
 company change of motion, change of position, action of 
 one body on another, and so on. Thus if a body in motion 
 imparts to another some of its motion, as through collision, 
 it has less power than before to change the motion of other 
 bodies. A body raised above the surface of the earth 
 appears to have the power to accelerate its own motion 
 towards the earth, or to move other bodies, and it gains in 
 this power as it is raised higher. A hot body has the power 
 of imparting heat to other bodies, and thus of producing 
 such effects as moving a railway train. These few instances 
 
Chap. I.] SUBSTANCE, MATTER. 7 
 
 illustrate not only the change in amount of power resident 
 on a body, but also the possibility of its transference from 
 one to another. 
 
 Further, it appears that under the same conditions a 
 •given body can or does take up always the same amount of 
 a given power. Thus a body moving with the same velocity 
 at different times, possesses the same amount of power to 
 move other bodies. A body acquires the same amount of 
 power in being raised through the same vertical distance at 
 different times. A stated body always emits the same 
 amount of heat in cooling through the same temperature 
 interval. These examples, beiug in advance of the present 
 stage of our subject, are cited only for the sake of adding a 
 touch of concreteness to the idea of " capacity " next to be 
 introduced. 
 
 Ability to receive, carry, or hold the " powers " is implied in 
 the definition of substance, — that is, a receptivity or capa- 
 city. But the cases just noted show that besides possessing 
 capacity in this abstract sense, each portion of substance 
 has a quantitatively definite and limited capacity for each 
 (although perhaps not for every) power. The term capacity 
 will be employed primarily with this meaning. There are 
 also other characteristics of substance commonly designated 
 as capacities, but they are all related to the powers in ways 
 which lead to no confusion with this primary denotation.' 
 All the capacities of substance are included under the 
 general name properties. This term is also used with 
 much lack of uniformity to include some or all of the 
 powers. 
 
 Substances, Elements. — Different portions of substance 
 exhibit more or less diverse groups of capacities. The 
 
 1 By the denotation of a term is meant the object, idea, or concept, 
 which the term replaces, that is, of which it is the name. By the connota- 
 tion of a term is meant the attributes of that object, idea, concept, which 
 are not included in the denotation hut which are brought before the mind 
 by the use of the term. 
 
8 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 capacities of a designated portion are to some extent per- 
 manent, and therefore characteristic of that portion. Thus 
 arises the suggestion that there are different substances, or 
 substances of different kinds. The number of such different 
 substances is very great, but each one differs from the one 
 most like it by an abrupt and pronounced change in proper- 
 ties ; in other words, the substances do not merge into one 
 another by imperceptible gradations. 
 
 By physical and chemical processes, there can be separated 
 or derived from all these substances a limited number, now 
 about seventy, whose properties are of a notably simpler 
 order than those of the majority. The most pronounced 
 point of distinction between these elementary substances 
 or elements, as they are called, and the others, lies in the 
 fact that all known processes fail to separate them into 
 simpler components. The elements have in common with 
 each other and with all other substances certain properties, 
 such as capacity for power through motion, capacity for 
 gravitation, etc. The characteristic differences between 
 them reside in the differences in their specific capacities 
 for certain of the powers, notably for the chemical, mag- 
 netic, electrical, cohesive, elastic, and heat powers. The 
 contrasts in these respects are great and seem at first sight 
 random, but the properties of the various elements prove 
 to be related by a very general though imperfectly appre- 
 hended law. Out of these elements all other substances 
 are constituted, either through mechanical mixture or by 
 chemical combination. 
 
 Those properties of the elements called capacities are 
 permanent, if by this it be understood that under the same 
 conditions a designated portion of a given element will 
 exhibit the same capacities. By conditions are meant 
 position and motion relatively to other bodies, and the 
 action exerted by those bodies, or by their powers at the 
 time. Laying aside consideration of the powers derived 
 
Chap. I.] SUBSTANCE, MATTER. 9 
 
 from motion and from gravitation as being common to all 
 substances, the quantities of other powers simultaneously- 
 present in a given portion of a given element vary from 
 time to time, as already remarked. This is obvious enough 
 as to heat, elastic power, and perhaps others. Some of the 
 capacities also differ in the degree or intensity of their 
 manifestation according to conditions, but these changes, 
 however great, constitute a change of state or of condition 
 of the substance, not a change of kind of substance. The 
 contrast between the same substance in different states is 
 so radical that regard to only a portion of the properties 
 would lead to the supposition of separate substances; as 
 for instance ice, water, and steam, or the solid, liquid, and 
 gaseous states of any of the elements. The chemical prop- 
 erties, and the methods of chemical investigation, are the 
 chief means of identification of substances. This arises 
 from the relative constancy of these properties, and from 
 the delicacy and simplicity of the methods. When chemi- 
 cal properties are under consideration to the exclusion of 
 ' physical, the term chemical substatice is sometimes employed. 
 No restriction or assumption has thus far been made or 
 is to be made concerning the nature of capacities. It is 
 possible, on the one hand, that the capacities are not 
 resolvable into anything more fimdamental. On the other 
 hand, they may prove to arise from so simple a thing as a 
 mode of motion of an ulterior su.bstance of simpler proper- 
 ties. No proof of the latter view exists, and no assumption 
 of its truth or probability will be adopted, but its possibility 
 cannot be ignored in forming fundamental concepts. 
 
 Throughout physics, and especially at the outset, a term 
 is needed which shall include as a part of its denotation all 
 the properties called capacities. To this use no other word 
 seems so well adapted, both by etymology and through the 
 sanction of custom, as substance; the singular term being 
 used in the abstract sense, and both singular and plural in 
 
10 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 tlie concrete. The question of whether the powers are 
 capable of existence separated from substance does not 
 affect the concept of substance. As to this point, it may be 
 here remarked that no one of these has been recognized 
 except in association with some portion of substance pos- 
 sessing properties other than those related to this power 
 alone. The case of radiation must be left to the paragraph 
 on the ether. 
 
 Matter. — Influenced by the considerations just presented, 
 but still more by others which can be offered only at a later 
 stage of the general subject, we are accustomed to form a 
 further concept underlying that of substance ; namely, that 
 of matter. Its presentation might more logically be de- 
 ferred, but convenience in the development of certain con- 
 trasts as we proceed renders desirable the introduction of 
 its definition here. 
 
 Matter is the Inert Constituent of Substance. 
 
 By inert is meant intrinsically devoid of power of any 
 kind. The more precise significance of inert must be left 
 to the chapter to which the definition of matter strictly 
 belongs. To that place may also be left the full discrimi- 
 nation between matter and substance. Briefly, however, we 
 may point out that if the capacities of a substance prove 
 ultimately not to be resolvable, then the concepts of matter 
 and substance become coextensive. If on the other hand, 
 they should prove to be resolvable into an effect or mani- 
 festation of " power " of some sort, or of a mode of motion, 
 impressed upon an inert substance or carrier, then matter 
 and substance would be distinct. The maintenance of this 
 distinction in dealing with both experimental and theoretical 
 physics will be found most helpful, particularly as tending 
 to preserve that freedom from bias which is essential to 
 sound scientific thought. While thus reasserting that we 
 
Chap. I.] SUBSTANCE, MATTER. 11 
 
 have here to do with an open question, we may point to the 
 theories of matter in Part II. as attesting the tenability of 
 the view that the distinction is not a mere fancy or ab- 
 straction. 
 
 Mass. — All views of matter assign to it the possession of 
 quantity. The term mass is used to denote this quantity, 
 so that. 
 
 Mass is Quantity of Mattes. 
 
 The term is used to express either the abstract idea or the 
 numerical quantity. 
 
 Weightal. — Quantity of substance as measured by "weigh- 
 ing " with the equal-arm balance will be denoted by the term 
 weightal, which is discussed at pages 111 and 152. Tlie term 
 is inserted here merely for the reasons of convenience stated 
 above respecting the term matter, and will not be employed 
 in advance of its further explanation in a way to compro- 
 mise the logical sequence of the main definitions. 
 
CHAPTER 11. 
 MOTION. 
 
 Definitions. — It will be necessary to employ certain terms 
 respecting motion, and to avoid misapprehension these will 
 now be defined. In the way in which they are here phrased 
 they belong to kinematics, that is, to the mathematical study 
 of pure motion as distinct from the physical study of the 
 motion of material bodies. 
 
 Position. — The position of a point may be determined by 
 its distance from at least three other points, or it may be 
 expressed by its distance from a single reference point and 
 the angles between a straight line drawn to it from the ref- 
 erence point and three other straight lines not all in the 
 same plane. These angles fix the direction point from the 
 reference point. Since there are no fixed lines or points in 
 space the choice of all reference points and lines is arbitrary, 
 and position is relative. 
 
 Motion is change of relative position. Motion is thus 
 purely relative and the motion of a particle will appear 
 different according to the reference point chosen. In de- 
 scribing motions we usually prefer to select for reference 
 such points as appear to remain sensibly fixed with refer- 
 ence to each other during the time-interval dealt with. 
 
 Path. — The successive points through which a particle 
 moves constitute its path. This may be straight, curved, or 
 irregular to any degree. 
 
 The /Space or Distance traversed by the particle is, unless 
 otherwise specified, the distance passed over, measured along 
 the path. 
 
 12 
 
Chap. II.] MOTION. 13 
 
 Direction. — If a particle is moved, towards a reference 
 point so that its successive positions lie along a straight line, 
 then that line indicates the direction of motion or displace- 
 ment of the particle. Motion is accounted positive when it 
 is towards one end of the line, and negative or in the opposite 
 direction when towards the other end. The particle is said 
 to be moved or displaced along the line, or in the direction 
 of the line, or towards the reference point, or in the direc- 
 tion of the reference point. In dealing with motion in 
 space, it is customary to refer the direction of motion to an 
 infinitely distant point unless otherwise specified. Though- 
 out a limited portion of space, all straight lines, whose 
 direction is towards the same infinitely distant reference 
 point, will be parallel, and will have the same direction. 
 Throughout any such finite space, therefore, any particles 
 moving along parallel straight lines will move in the same 
 direction. Thus, in the general sense of the term, a particle 
 is said to be moving in the direction of a specified straight 
 line when it is moving either along that line or along any 
 line parallel to it. 
 
 Still more generally a particle is said to be moving towards 
 (or away from) a reference point, when its distance from 
 that point is lessening (or increasing), however irregular or 
 indirect may be the actual line of motion. In this case it 
 is also said to be displaced towards (or from) the reference 
 point. Or, if the direction is specified by means of a refer- 
 ence line, instead of the point, the particle is said to be 
 moving in the direction of the line or to be undergoing 
 displacement in that direction so long as its distance from a 
 point infinitely distant along the line is continually lessen- 
 ing, whatever may be the actual course or path of the particle. 
 
 The direction of motion of a particle along its path is 
 the direction of the path at the given time. If the path is 
 not straight, this direction is that of its tangent at a speci- 
 fied point. 
 
14 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 Velocity is the time-rate of motion (expressed, for in- 
 stance, in feet per second or meters per second). If a par- 
 ticle moves over equal spaces in equal times, however long 
 or short, its velocity is constant; if the spaces are suc- 
 cessively larger, the velocity is increasing ; if smaller, 
 diminishing. The motion in these three cases is called 
 respectively uniform, accelerated, and retarded motion. 
 
 The Average Velocity through any given time-interval is 
 the space traversed in that time, divided by the time. If 
 the motion is uniform, the average velocity is constant, how- 
 ever long or short the time-interval. 
 
 Instantaneous Velocity. — If the motion of the particle is 
 varying, its velocity at any given instant is the limiting 
 value of the average velocity during a short interval of 
 time as this interval is taken smaller and smaller. If the 
 motion is uniform, the instantaneous velocity at all times is, 
 of course, identical with the average velocity. In general 
 when the term velocity is used, instantaneous velocity is 
 implied. 
 
 Rest. — This is merely a special case of motion in which 
 the velocity relative to the chosen reference point is zero. 
 Any particle will thus be at rest, vrhatever its motion relar 
 tive to other points, when referred to a reference point 
 possessed of the same motion relative to those points. 
 
 Acceleration is any increase of velocity. Acceleration 
 may be continuous or sudden. Thus the velocity may be 
 continuously increasing or may undergo increase for a brief 
 time only, again becoming uniform ; or may receive a suc- 
 cession of such impulses. Since all motion is relative, an 
 increase of velocity differs in no respect whether the par- 
 ticle is initially at rest or in motion, so that the motion of a 
 particle started from a state of rest by a sudden impulse 
 and then continuing to move uniformly, is as much an ac- 
 celeration as though the particle were originally moving at 
 some uniform velocity. 
 
Chap. II.] MOTION. 15 
 
 Toted Acceleration is the total increment of velocity in any 
 stated time. 
 
 Sate of Acceleration is the time-rate of increase of velo- 
 city. The words rate of are usually omitted in speaking of 
 this quantity. If this time-rate is constant, the motion is 
 uniformly accelerated. 
 
 Retardation. — If the spaces traversed in successive, equal 
 intervals of time are diminishing, this diminution of velo- 
 city is called retardation. 
 
 The State of Motion, or simply the motion, of a particle is 
 the instantaneous velocity and direction of motion referred to 
 any chosen point and line. 
 
 Change of State of Motion. — Any change of motion or of 
 state of motion is therefore a change in either instantane- 
 ous velocity or direction or both, as thus referred. Thus, 
 change of motion may be either acceleration, or retardar 
 tion, or change of direction alone, or either acceleration or 
 retardation combined with change of direction, depending 
 upon the reference point chosen. 
 
 The state of motion of a particle at a stated point of its 
 path would be shown by the motion of an imaginary point 
 moving with the instantaneous velocity of the particle and 
 in the straight line tangent to the path. The motion of a 
 particle might be referred to a point moving in this manner 
 beginning at any chosen instant. A little reflection will 
 show that any change of motion whatever will appear, 
 when thus referred, merely as an acceleration of the parti- 
 cle in some direction starting from a state of rest. Any 
 change of motion results therefore from the superposition of an 
 acceleration as thus referred in a suitable direction upon some 
 previous motion. The acceleration may be continuous or an 
 impulse. 
 
 Thus if a particle were moving uniformly in a straight 
 line with regard to the chosen point, an acceleration in the 
 direction opposed to this motion would produce retardation 
 
16 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 or retarded motion. An acceleration in the direction of 
 motion would produce accelerated motion. Acceleration at 
 an angle to the direction of motion would produce change 
 of direction and in general also change of velocity, but 
 if in a direction always at right angles to the path, it 
 would produce only change of direction. 
 
 Acceleration: Another Usage. — The term acceleration is 
 often employed in abstract dynamics to include both re- 
 tardation and change of direction as well as acceleration as 
 above defined. In that usage positive and negative acceler- 
 ation correspond respectively to acceleration and retarda- 
 tion as above defined. Following such usage all change of 
 the state of motion would be called acceleration. This 
 notation is discussed in Maxwell's " Matter and Motion " 
 [London and New York, 1882, 18mo, p. 128] which should 
 be read by students of this subject. 
 
 The terms liind or mode of motion are to be distinguished 
 from state of motion. They refer to the motion which a 
 particle, or system of particles, has been undergoing or will 
 undergo ; that is, they refer to the past and future motion 
 of a particle and not simply to its instantaneous motion. 
 Thus the motion of a particle in a circle would be a kind or 
 mode of motion, not a state of motion. 
 
 Two modes of motion to which reference will be made 
 are translatory and rotary motion. A particle, or a body 
 of sensible size, is in translatory motion when the paths of 
 all portions of it are parallel straight lines. A particle, or 
 a body of sensible size, is in rotary motion when each point 
 of it is moving in a circular path about some imaginary axis. 
 The two motions may be, and usually are, combined in the 
 motion of any actual object. 
 
 Free Motion. — This term belongs to dynamics and not to 
 kinematics. It is the motion of a body when not restrained 
 in any direction, that is, when not interfered with by the 
 power of another body (p. 3). 
 
Chap. II.] MOTION. IT 
 
 A body may be free to move in any direction, even when 
 subject to any number of external actions, provided the 
 effects of these actions balance one another and remain 
 balanced in vv^hatever direction the body may be m.oved. 
 The conditions necessary for perfectly free motion are diffi- 
 cult to fulfil in practice, but nearly perfect freedom along 
 a line or in a plane may readily be obtained. Thus a ball 
 sliding upon smooth ice has nearly perfect freedom of mo- 
 tion in a horizontal plane ; a billiard ball has some such 
 freedom upon the table ; a ball upon the bowling alley is 
 also nearly free in a horizontal plane, or a carriage rolling 
 upon a smooth level floor. 
 
CHAPTER III. 
 ENERGY. 
 
 Laws of Motion. — The most casual observation shows us 
 that, under ordinary conditions, bodies set in motion come 
 to rest sooner or later, unless some effort is exerted to main- 
 tain their motion. To ascribe to motion a tendency to de- 
 cay is therefore most natural, and this was long done. 
 Little further observation however is required to show that 
 the rate at which the motion of any given body diminishes, 
 depends upon the conditions among which the body moves. 
 Thus, if started each time with the same velocity, a ball 
 would move but a short distance over a pebble-covered road, 
 further over a good road, much further still over a smooth 
 floor or over ice, and still further if free in air. It is obvi- 
 ous that the irregularities which constitute the roughness 
 of these successive surfaces oppose the motion of the ball. 
 Also, opposition is afforded by the air, as we know from 
 moving the hand rapidly through it. In these eases, then, 
 the less the opposition, the longer the motion persists. By 
 the multiplication of such evidence, confirmed by studies 
 of the motion of celestial bodies, and by general experience, 
 we arrive at Newton's first law of motion. 
 
 Using the terms with the meanings already assigned to 
 them, this law may be expressed thus : — 
 
 The state of motion of any body remains constant, except in 
 so far as changed by external action. 
 
 It is to be noted that this law is not a direct statement of 
 observed fact, but an inference as to what the motion of a 
 free body would be, based on the observed motions of bodies 
 
 18 
 
Chap. III.] ENERGY. 19 
 
 which are not free. No actual body is found which is en- 
 tirely free. 
 
 A more direct general statement of the observed facts 
 may be put in the form that : — 
 
 All observed changes in the state of motion of bodies are due 
 to discoverable external action. 
 
 Throughout these statements the body considered is as- 
 sumed to be permanent as to the relative position of its 
 parts. It is easy to show later, that what is here referred 
 to as the motion of the body, is the motion of a point occu- 
 pying a definite position relatively to all portions of the 
 body, and that the above law applies to the motion of that 
 point whether the parts of the body be in motion relatively 
 to each other or not. 
 
 Inertia. — The doctrine or principle, less correctly called 
 the property, of inertia, is implied in Newton's first law of 
 motion. It may be more explicitly stated thus : ■ — ■ 
 
 No body has poicer to change its own state of motion. 
 
 The doctrine or property of inertia is a purely negative 
 one, a mere denial of the possession of a certain power by 
 bodies. It does not ascribe to bodies or to substances any 
 property in virtue of which external action is required to 
 change their state of motion, or in virtue of which they 
 possess when in motion any power relatively to other 
 bodies ; although such a character is often ascribed to this 
 property. It is of course understood that this expression 
 does not deny to a body the power of relative change of 
 position of its parts. It refers merely to the relation of the 
 body as a whole to external objects. 
 
 Definition of Energy. — We now arrive at the most con- 
 spicuous and important of modern physical concepts ; that 
 of Energy. 
 
 By Newton's first law of motion, just stated, any change in 
 the state of motion of any body, that is, any deviation from 
 uniform velocity of motion in a straight line, is due to exter- 
 
20 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 nal action. To the jiower to exert this action, is given, in all 
 physical science, the name, Energy. We may define the term 
 thus : — 
 
 Energy is Power to Change the State of Motion 
 OP A Body. 
 
 The motion i^ to be referred to any point not involved in 
 the change of motion to be observed. This is the most gen- 
 eral mode of reference possible, and therefore the best for 
 the purpose of the definition. 
 
 Energy stands to change of state of motion as cause to 
 effect. Without energy there can be no change in state of 
 motion ; and consequently, vsrherever change in state of mo- 
 tion is in progress, there energy must be in action. Such 
 change affords indisputable and sufficient evidence of the 
 presence of energy in any specific case. 
 
 The form of this definition is a departure from prevailing 
 usage, as shown in Chapter XII., with the reasons therefore. 
 Its merits or defects must be developed through its applica- 
 tion to all physical phenomena and knowledge, and test of 
 its value will thus be afforded at nearly every subsequent 
 page of this book. Should it fail to discriminate correctly 
 in every case between that which is energy, and that which 
 is not, as known by other reliable tests (if such there are), 
 it would thereby be condemned. If it be found defective 
 in logic, it must be abandoned. If it be found inconvenient 
 or difficult of comprehension, then it should be replaced by 
 a better. 
 
 The definition may, at first sight, appear insufficient, as 
 energy manifests itself to us in such a multitude of diverse 
 ways, most of which we do not recognize as due to motion 
 at all, and many of which are not known to arise from mo- 
 tion. Thus, energy produces in us the sensations of light, 
 color, taste, smell, sound, warmth, and so on. It gives rise to 
 all the phenomena of Nature around us. Still, we are un- 
 
Chap. III.] ENEBGY. 21 
 
 able to trace the nature of the energy in most of these oc- 
 currences to even the limited extent of ascertaining whether 
 it is or is not due to motion of some sort. In a few classes 
 of cases only do we know that it is so. But we do know 
 that energy, in whatever form it manifests itself, can be 
 made to produce change in state of motion of bodies. In 
 other words, it can be changed into energy of motion of a 
 material body. Proofs, or rather illustrations of this, will 
 be given in the following chapter. Further, energy of mo- 
 tion is the sole form of energy of whose nature we have 
 adequate knowledge. Again, the definition arrived at 
 through energy of motion, and advanced above, appears to 
 be more simple, and more easily grasped by the beginner, 
 than those iisually employed. It also renders possible the 
 introduction of the subject of energy at the very outset of 
 the study of physics with entire logical consistency. These 
 and other considerations, some of which appear elsewhere 
 in due course, together with its ascertained effectiveness in 
 developing the other concepts of physics, have decided the 
 author in its adoption. Without further present comment 
 on this definition, we shall proceed at once to its applications. 
 In dealing with the motion of actual bodies with reference 
 to energy, the selection of the reference point for the motion 
 is often of considerable importance. In the foregoing gen- 
 eral definition the motion is referred to any point whatever. 
 In any particular case we are concerned with the energy of 
 motion of only a limited number of bodies. Such a set of 
 bodies may be called a system. In any such system we are 
 ordinarily concerned with the motion of one or more bodies 
 relatively to the system as a whole. The state of motion of 
 these bodies would then be their motion relatively to the sys- 
 tem as a whole, or relatively to some point at rest with re- 
 spect to the system, that is, possessing relatively to any other 
 body the same motion as the whole system. Some little care 
 is frequently necessary in the selection of such a reference 
 
22 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 point. In dealing with objects near the earth's surface, 
 this forms a convenient reference point since such systems 
 of bodies would generally be at rest with reference to this 
 surface, or would possess some known motion with regard 
 to it. In a system in motion relatively to the earth's sur- 
 face, we frequently select some member of the system which 
 for sufficient reasons we know to be not involved in the 
 changes of motion with which we have to deal. The most 
 general of all physical reference points and directions are 
 of course those of the fixed stars. 
 
 Muscular Energy. — As far as concerns the continuity of 
 the set of physical definitions presented in this book, all 
 that is given under this heading, " Muscular Energy," might 
 be omitted. The purpose in retaining it is merely to indi- 
 cate an easy transition from ideas familiar to almost every 
 one to a more abstract and purely scientific definition. As 
 will be again noted, however, this definition is in no wise 
 dependent upon, or restricted by, this introduction. 
 
 Whatever point be chosen for reference, the motion of 
 bodies is in general imdergoing change. If now by our 
 own action we produce in moving bodies or in those at 
 rest, any change of motion which they would not otherwise 
 undergo, we must exert muscular effort. That we must do 
 so follows directly from the first law of motion (p. 18). 
 We also become aware of it through direct experience. For 
 the power or capacity to exert such muscular effort we have a 
 well-recognized term, muscular energy, or energy. 
 
 Exercise of the mind or nervous system accompanies 
 muscular effort, but is directive of, rather than a part of, 
 the muscular energy. 
 
 The term muxcidar energy or energy as thus defined con- 
 veys to every one an entirely familiar and sufficiently definite 
 conception. This and the further fact that energy is used 
 in much the same sense with reference to inanimate objects 
 in physics, renders this conception a quite suitable starting- 
 
Chap. III.] ENERGY. 23 
 
 point from wliich to approach the physical concept. The 
 purely physical definition is in no way based upon or 
 derived from the definition of muscular energy, so that any 
 want of precision in the latter does not vitiate the scien- 
 tific definition. The notion of muscular energy, however, 
 gives concreteness or tangibility to the otherwise abstract 
 idea, — a consideration of the utmost importance in the 
 elementary presentation of the subject. It may here be 
 noted, however, that although the sensation accompanying 
 muscular effort is in no way directly involved in the physi- 
 cal definition, yet it forms an ineradicable part of the conno- 
 tation (foot-note, p. 7) of the term, and exerts a very 
 considerable, perhaps not altogether wholesome, influence 
 on certain other physical concepts, as will be later indicated. 
 
 We have been dealing with the general case of the motion 
 of any body, and therefore with the motion of a body not 
 entirely free. The effort producing change of motion of 
 such a body is expended in two ways. One portion goes to 
 move the object against the obstruction to its free motion. 
 The other part is exerted in merely accelerating or otherwise 
 changing the state of free motion of the object. 
 
 That muscular effort — and hence energy — is necessary in 
 changing the motion of a free body not only follows from 
 the converse of Newton's first law, but is easily illustrated 
 by familiar experience. For instance, to throw a ball hori- 
 zontally, that is to set it in rapid motion, requires a consid- 
 erable effort. To maiatain the speed of the ball, however, 
 obviously calls for very little effort, since it will proceed a 
 long distance with no application of effort whatever from 
 without. The effort of throwing the ball is mainly ex- 
 pended in the mere acceleration of its motion. A large 
 door swinging on well-oiled hinges can be easily kept moving 
 at any desired speed after that velocity has been reached, 
 but to bring its speed up to that point requires a distinctly 
 greater effort. To roll a carriage over a smooth level floor, 
 
24 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 a car upon smooth level rails, a loaded sled on. ice, all afford 
 examples of effort expended in mere acceleration of bodies. 
 In all of these it is easily perceivable not only that effort is 
 expended in acceleration but that the amount of effort de- 
 pends upon the rate of acceleration, that is upon the time 
 in which a definite increase of velocity is produced, the 
 effort being measured by the push or pull exerted — terms 
 whose ordinary meaning is sufficiently definite for the pres- 
 ent purpose. 
 
 The introduction of the term energy, by means of its 
 familiar association with muscular effort, might usefully 
 precede the scientific definition of the term in the ele- 
 mentary presentation of the subject. 
 
CHAPTER IV. 
 FORMS OF ENERGY. 
 
 Recognition of Forms of Energy. — Having thus postulated 
 a definition of energy, we proceed to ascertain in what forms 
 energy presents itself. The definition affords us a test for 
 energy. Wherever change in state of motion occurs, energy 
 must be in action, whether or not we are able to determine 
 in just what the energy consists or precisely where it resides 
 in the given instance. If the definition is adequate, this 
 test will never lead to the mistaking of anything else for 
 energy, or ultimately to the overlooking of any form of 
 energy. 
 
 The disclosure of all the ways or forms in which energy 
 manifests itself demands ultimately the examination of all 
 the phenomena of the universe. Happily, however, this in- 
 finite task is reduced to one of limited magnitude through 
 the classification of phenomena into a few great groups such 
 as the phenomena of cohesion, of gravitation, of elasticity, 
 etc. Each of these groups then corresponds to a property 
 (of the kind which is here called a " power " ) of bodies or 
 substances. And as we apply to each the test for energy, 
 we shall find that each is due to some manifestation, or as it 
 is called some form of energy. How far these apparently 
 different forms are in their real nature identical we must sub- 
 sequently discuss. We thus arrive at a system of nomen- 
 clature for the forms of energy which, however empirical, 
 is so interwoven with colloquial usage and with the histori- 
 cal terminology of physics that it must long survive. 
 
 Kinetic Energy. — It is an observation of the most familiar 
 
 2i? 
 
26 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 kind that a body in motion relatively to another possesses 
 power to change the state of the other's motion by collision. 
 The first body therefore possesses energy with reference to 
 the second. It is also easy to discover that the amount of 
 the power possessed by the body depends upon its velocity 
 relatively to the second. But in ordinary cases of collision, 
 so many incidental actions of different kinds occur that the 
 conditions must be much simplified to demonstrate the law 
 of this energy. 
 
 Take two ivory balls A and B, of equal diameters and as 
 nearly as possible alike in all respects. Suspend each by a 
 long silk thread, or better by two threads diverging upward. 
 Place each ball so that when hanging free the two just touch 
 each other, and so that their point of contact is in the line of 
 free swing of either. The two balls will then have nearly per- 
 fect freedom of motion when moving in a horizontal plane 
 through their point of contact, or at least in one direction in 
 that plane. Let B remain at rest, and draw A to one side 
 and release it. Just before reaching B, it will have ac- 
 quired a definite velocity. On colliding with B, A will 
 come completely to rest (or sensibly so), and B will move 
 off with a considerable velocity. The motion of B has thus 
 undergone acceleration. By the definition then, energy has 
 been exerted upon it. In what did that energy consist ? 
 
 By close scrutiny of the conditions of the experiment we 
 can discover, beyond the changes of velocity of A and B, no 
 other changes except certain minor ones. The latter consist 
 chiefly of vibrations of the parts of the two balls and of the 
 surrounding air, and of a minute amount of heat developed 
 in the balls. These minor changes may be disregarded at 
 present as unimportant. Subsequent investigation of them 
 would show that due allowance for them only strengthens 
 the inference here drawn. During the interval occupied 
 by the collision, the only sensible change was the loss 
 of velocity by A and the acquisition of velocity by B. 
 
Chap. IV.] FOBMS OF ENERGY. 27 
 
 A then must have possessed energy with reference to B. 
 Also it must have possessed this energy in virtue, partly at 
 least, of its velocity with reference to B. 
 
 If A is given a higher velocity, the acceleration of B will 
 be greater; so that the energy of A appears to increase 
 when the velocity is increased. 
 
 If a larger ball of the same substance be used instead of 
 A and be given the same velocity as A, it will impart to B 
 a different acceleration. If balls of different substances be 
 used, differences in the acceleration produced will be ob- 
 served. But most substances are imperfectly elastic so that 
 in the collision of such bodies the incidental changes, such 
 as crushing, production of heat, etc., are not negligible. 
 
 From the foregoing, however, we are justified in conclud- 
 ing that a body in motion possesses energy in virtue jointly 
 of that motion, and of some property of the body itself. 
 This form of energy is called energy of motion or kinetic 
 energy. 
 
 The foregoing inference is confirmed by universal experi- 
 ence with bodies and substances of every sort. The law 
 of kinetic energy will be demonstrated in a later chapter, 
 and a more general method of illustration will be described. 
 
 Kinetic Energy of Rotation. — If a body be in rotation 
 about a definite axis, each point or particle of the body has 
 at any given instant a definite direction and velocity of 
 motion. The direction is that of the tangent to the path 
 which it is describing, and its velocity is the instantaneous 
 velocity. The body may therefore be regarded as made up 
 of an indefinite number of very small portions, each of 
 which has its own velocity and therefore its own amount of 
 kinetic energy. The sum of all these amounts will be the 
 total kinetic energy of rotation of the body. 
 
 Molar, Molecular, Atomic Kinetic Energy. — A distinction 
 is convenient between the kinetic energy possessed by a 
 single molecule, and that possessed by an aggregate of 
 
28 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 molecules, — admitting for the moment the molecular hy- 
 pothesis of the constitution of bodies. The energy pos- 
 sessed by a single molecule in virtue of its velocity is 
 called molecular kinetic energy. That possessed by a body 
 composed of more than one molecule is called molar kinetic 
 energy, and similarly we may have atomic kinetic energy. 
 These may be due either to motion of translation or to 
 rotation. 
 
 Gravitation Energy. — One of the most familiar phenomena 
 exhibited by bodies may be stated in this way: Every 
 body tends to approach every other body. In other words, 
 any two bodies iviU approach each other unless prevented 
 from so doing by some external means. 
 
 This at first sight appears directly contradictory to the 
 doctrine of inertia, but if we adopt the views above pre- 
 sented with regard to inertia and energy, and pursue the 
 line of inquiry just enunciated, this contradiction at once 
 disappears. If analyzed, the above statement will be found 
 to mean that each of any two bodies, if allowed entire 
 freedom of motion, has its motion accelerated relatively to 
 its previous state of motion ; that is, the state of motion of 
 the body is changed. Therefore, according to the criterion 
 which we have imposed for energy, there must be energy 
 of some sort in action, causing the change of the motion of 
 these bodies. Here then we have directly an evidence of 
 the existence of energy. 
 
 What is the character of this energy which causes gravi- 
 tation ? To this no satisfactory answer can be given, but 
 certain attempted explanations are presented in a later 
 section. There is no discoverable change in the bodies 
 concerned, or in any of their surroundings, which takes 
 place at the same time and in the same measure as 
 the change of motion of the bodies. There is no direct 
 evidence that any body or bodies whatever are producing 
 this change through collision with the bodies concerned; 
 
Chap. IV.] FORMS OF ENERGY. 29 
 
 that is, there is no evidence that this action of gravitation 
 is due to any kinetic energy. Inspection fails to show 
 whether the energy resides upon the bodies or is external 
 to them, and has failed also, thus far, to give a thoroughly 
 satisfactory hypothesis as to its nature. We are, however, 
 in no way relieved by this fact from the inference that this 
 energy exists ; and this conclusion leads us into no contra- 
 dictory positions, but on the other hand aids in establishing 
 a simple and satisfactory system of ideas. 
 
 We thus have demonstrated the existence of energy asso- 
 ciated with substances in a way which is different from 
 kinetic energy, or at least which we are unable as yet to 
 demonstrate to be due to kinetic energy. We are therefore 
 obliged to content ourselves with saying that this is another 
 form of energy. 
 
 Briefly, then, our second form of energy discovered, is 
 that which causes the phenomena of gravitation. 
 
 Heat. — We know familiarly that through various ma- 
 chinery, such as a steam engine, etc., we are able to set 
 objects in motion, and we know that the power to produce 
 this result is derived from the heat which is generated 
 under the boiler of the steam engine. We have, then, 
 another case in which the motion of a body is changed, and 
 in this case we are able to go a step farther than in gravita- 
 tion, inasmuch as we can say that this energy is identical 
 with something with which we are familiar in other ways, 
 namely, heat, — whatever the mechanical nature of heat 
 may prove to be. The above illustration is, of course, not a 
 rigid proof ; but a perfectly strict demonstration that heat 
 is convertible into molar kinetic energy is easily possible. 
 
 Heat, then, is still another form of energy ; although the 
 nature of heat energy is a matter of hypothesis. If we 
 admit the molecular constitution of bodies, then it is easily 
 shown that heat is probably due to the vibratory motion of 
 the molecules within a body, and that it is in reality the 
 
30 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 total kinetic energy of vibration of the molecules apart from 
 any translatory energy of the body as a whole. 
 
 Energy of Elasticity. — If a bent spring be released, the 
 substance of the spring is set in motion, that is, its state of 
 motion is changed. Or if the spring, when released, be 
 allowed to act upon a body separate from itself, the motion 
 of that body will be accelerated. Some form of energy must, 
 therefore, be here in action. The change which we recognize 
 as taking place concomitantly with this exhibition of en- 
 ergy is a change of form of the spring. Some portions of its 
 substance are extended beyond their normal length while 
 others are compressed. This change of form cannot be the 
 energy, but merely indicates that the energy is brought into 
 action through change of relative position of parts of the 
 substance of the spring. Here again we are entirely igno- 
 rant as to the mechanism of the form of energy. This 
 property of bodies, which is called elasticity, is somewhat 
 complex in its manifestations and possibly in its nature ; but 
 without attempting to analyze the phenomena thoroughly, 
 we are evidently at liberty now to say that wherever the 
 property of elasticity is in action, a form of energy is pres- 
 ent. This form may be called the energy of elasticity, or 
 may be given the not very euphonious name, elastic energy. 
 From the fact that this energy is always brought into action 
 through the deformation of the body, we might infer that 
 the energy was resident upon the body; but the most that 
 can be positively affirmed, in the case of solids and liquids 
 at least, is that it is intimately associated with them. A 
 fundamental theory of the nature of elasticity is stated 
 later, in the section on the Vortex-atom Theory. A theory 
 explaining the molar elasticity of gases is stated under the 
 Kinetic Theory of Gases. 
 
 Cohesion Energy. — "When we attempt to pull apart any 
 two portions of a solid body, we find that the body offers 
 resistance, and that the pull must exceed a definite amount 
 
Chap. IV.] FOBMS OF ENERGY. 31 
 
 to break the object. Again a small portion of any liquid, if 
 free from external disturbances, will assume a spherical 
 form. A rain-drop is nearly spherical. A drop of water or 
 mercury upon a horizontal surface which it does not wet 
 assumes the shape of a flattened sphere, the flattening being 
 due to the interference of gravitation. These phenomena 
 and others analogous to them indicate that the particles of 
 a body, when in close proximity, tend to . approach one 
 another, and will do so unless prevented by some external 
 action. 
 
 We know, from commonest experience, that the size of 
 bodies is due in part to the presence of heat in these bodies, 
 and that if some heat is withdrawn the bodies become 
 smaller, that is, their particles approach one another. This 
 approach is due to the property of cohesion and is a direct 
 evidence of the action of some form of energy, since it 
 involves a change in state of motion of the parts. The 
 motion of the parts of the drop of liquid in assuming the 
 spherical form under the action of cohesion is similarly 
 direct evidence of the action of energy. There is present, 
 therefore, some form of energy which causes the phenomena 
 grouped under the name cohesion. The phenomena of 
 elasticity when bodies are stretched, and those of cohesion, 
 are closely interwoven. But no embarrassment is introduced 
 by assuming that, distinct from energy of elasticity, there 
 is a form of energy which may be called the energy of co- 
 hesion. This form appears at first sight to resemble that 
 of gravitation, but its effect appears to vary with the in- 
 crease of distance between the particles much more rapidly 
 than the law of gravitation indicates. Certain explanations 
 of this apparent discrepancy, however, have given plausi- 
 bility to the view that cohesion is identical with gravitation 
 (cf. Part II.). 
 
 Chemical Energy. — In general, it is found that portions 
 of any two different elements show a tendency to combine 
 
32 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 with each, other, forming a compound with properties dif- 
 ferent from those of its constituents. These compounds 
 in turn manifest certain tendencies towards combination 
 or other chemical reaction with each other. The process of 
 combination is attended, in very many instances, with the 
 evolution of heat or with change of volume. The production 
 of heat in such a case where no external condition is changed, 
 may be regarded as evidence that the chemical action was 
 the result of the action of energy in some form; for this 
 heat may be transformed into molar kinetic energy, and 
 thus the chemical action is made to give evidence that it is 
 an action of energy. The fact that this process consists of 
 two steps, first, a production of heat ; second, a transforma- 
 tion of heat into molar energy, is not to be regarded as im- 
 pairing the validity of the inference. For our criterion for 
 determining what energy is, does not imply that the change 
 of state of motion must be produced directly. Indeed if 
 we analyze the very first instance which we have stated, 
 namely, acceleration of one elastic body by impact of an- 
 other, we find that this operation also is not a direct one. 
 The energy possessed by the moving body is entirely trans- 
 formed into energy of elasticity, and again re'transformed 
 into kinetic energy, during the impact. 
 
 In case of the change of volume during chemical combina- 
 tion, where this has not been brought about by any external 
 influence, this change is direct evidence of energy, under 
 our criterion. 
 
 There are many cases of chemical reaction where the ab- 
 sorption of heat, or of energy of some other form, from 
 without attends the operation. This energy disappears, 
 that is, no longer continues to exist in its original form, but 
 it may be traced, on further analysis, in the physical or 
 chemical changes which occur during the chemical action. 
 Without pursuing this into detail, it is clear that we are 
 justified in assuming that chemical action and chemical af- 
 
Chap. IV.] FORMS OF ENERGY. 33 
 
 finity, so called, are due to a form of energy which we will 
 call chemical energy. 
 
 A more vivid illustration of the existence of chemical 
 energy is in the action of explosives. Here the chemical 
 energy gives rise not only to violent change of motion but 
 to other actions by which we may recognize energy. 
 
 With the nature of this form of energy we are as little 
 acquainted as with that of gravitation or elasticity. But a 
 very remarkable stride in the direction of explanation has 
 been made under the Vortex-atom Theory, as later described. 
 - Electrical Energy. — Under certain conditions a conduct- 
 ing body, such as a metallic wire, exhibits properties which 
 we ascribe to a current of electricity flowing through the 
 wire. The conductor containing the current becomes heated 
 by the passage of the current, and by proper disposition of 
 other conductors about this one, motion may be set up, as 
 in electric motors and various other well-known forms of 
 electrical apparatus. These actions begin and end with the 
 current, and are clearly due either to it or to the same cause 
 which produces it, and by our criterion must be due to a 
 form of energy. This we call electrical energy. 
 
 This form of energy is manifested in another way. 
 Under certain well-known conditions, conductors of elec- 
 tricity when insulated from other conductors become 
 charged, as it is called, with electricity. Two such con- 
 ductors tend to approach or recede from one another ac- 
 cording to the character of the charges upoji them, that 
 is, they will so move unless prevented ; therefore these 
 charges are either due to or attended by a form of energy 
 which appears to be fundamentally identical with that 
 which attends currents of electricity. The difference in 
 the phenomena exhibited by electricity at rest upon charged 
 bodies, and electricity in motion in conductors, appears to 
 be due solely to the motion. We do not, for our present 
 purpose, need to examine further the many properties and 
 
34 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 phenomena of electrified bodies, since those noted are suf- 
 ficient to establish the existence of electrical energy. 
 
 Magnetic Energy. — The natural lodestone is capable of 
 moving a small mass of iron. The artificial magnet has 
 the same power. This power results from a condition 
 apparently attending and always associated with portions 
 of iron. It is also possessed in widely varying degrees by 
 nickel and some other substances and is called magnetism. 
 The closely related property known as diamagnetism is 
 possessed by a large number of substances. This power to 
 move bodies arises, therefore, from some form of energy 
 closely associated with the magnetic body. It may be 
 called magnetic energy. Of its nature we know, first, 
 that it is very closely related in some way to electrical 
 energy; second, that at present the most satisfactory ex- 
 planation of its action is upon the assumption that magnetic 
 energy and electrical energy are identical. 
 
 Energy of Sound Vibrations. : — We know that a body trans- 
 mitting sound is in a state of vibration. Its particles are 
 moving to and fro in a systematic manner, each particle 
 passing from a state of maximum velocity to a state of 
 zero velocity and back again at regular periods, the motion 
 being in the direction in which the sound is propagated. 
 When in this condition of maximum velocity, each particle 
 possesses a definite maximum of kinetic energy. As it 
 passes in its vibration towards its state of zero velocity, 
 it gradually yields this kinetic energy, which is trans- 
 formed into energy of elasticity of the body. This energy 
 of elasticity is in turn expended in accelerating the motion 
 of the particle, and the process is thus one of successive 
 transformation of kinetic into elastic energy and the re- 
 verse. The result is a production of a wave of energy 
 through the transmitting body. The total of all this 
 kinetic and elastic energy within the body may be called 
 the energy of sound contained by that body. It is not neces- 
 
Chap. IV.] FOBMS OF ENERGY. 35 
 
 sary to choose between the subjective and objective defini- 
 tions of sound, but for clearness we may say that the 
 sensation of sound originates in, and is transmitted by, the 
 vibratory energy of elastic substances, and that this vibra- 
 tory energy consists, at any given instant, partly (equally) 
 of kinetic and elastic energy in the substance. This resolu- 
 tion into kinetic and elastic energy is capable of rigid 
 proof, so that the energy of sound is not to be classed as 
 a separate form. 
 
 There are other known forms of vibration energy, such 
 as that of the swinging pendulum, of tlie planets in more 
 or less elliptical orbits, etc. Each of these consists, how- 
 ever, of a joint action of kinetic energy and gravitation 
 energy or some other form. They, like sound energy, are 
 therefore not to be classed as new forms. 
 
 Radiant Energy: Light, Radiant Heat, Electro-magnetic 
 Radiation. — It is a matter of observation that hot bodies 
 produce upon other objects in the vicinity certain effects, 
 even when entirely unconnected with them by any other 
 substance than that which can exist in the most perfect 
 vacuum attainable. The nature of the effect depends 
 upon the receiving object. Thus if the body be sufficiently 
 hot, the human eye is affected in such a way as to produce 
 the sensation which we call light. The blackened bulb of a 
 thermometer is so affected that the thermometer indicates 
 a rise of temperature, that is, indicates that heat has been 
 introduced into the bulb. If the receiving object is a sensi- 
 tive photographic plate, chemical action is promoted. Of 
 these three phenomena, one, namely the production of heat 
 in the bulb of the thermometer, affords evidence that by 
 whatever process the hot body may have produced its action 
 upon the thermometer, the essential part of the process 
 consisted in the transmission of energy of some form to the 
 thermometer ; for heat was produced in the bulb of the in- 
 strument. Now this heat can be made to accelerate the 
 
36 FUNDAMENTAL PHYSICAL CONCEPTS. [Paet I. 
 
 motion of a body ; therefore that which produced the heat 
 must have been a form of energy. This power of a hot 
 body is called radiation. Kadiation, therefore, in this case 
 is a process of transmitting energy in some way to objects 
 not in direct contact with the radiating body. Hypothesis 
 is necessary to explain the nature of this process of trans- 
 ference, but we need not enter into this explanation at the 
 present point since the only thing which we wish here to 
 establish is the existence of a form of energy by which the 
 transference is made ; and that has already been done. 
 
 This power of radiation is possessed undoubtedly, not 
 only by hot bodies, but by all bodies with temperatures 
 above the absolute zero; also by bodies in which rapid 
 oscillations, or rapid periodic changes of position, of elec- 
 trical charges, are taking place ; also by bodies in which 
 rapid magnetic changes are occurring; also by certain 
 vacuum tubes through which high-pressure electrical dis- 
 charges are taking place. The effects produced in these 
 various cases are not all the same, but they are all equally 
 capable of being proved to be due to a form of energy trans- 
 mitted from the radiating body to the receiving body. This 
 is called radiant energy, or at least this term may for the 
 present be held to cover it. It is supposed to be due to the 
 vibrations of the (hypothetical) luminiferous ether which is 
 supposed to be universally distributed through space. 
 
 Muscular Energy. — In a preliminary classification, it 
 would be necessary to include this title. But investiga- 
 tion shows that the energy of the muscles is apparently 
 not an independent form, but only a more or less complex 
 manifestation of some of the foregoing forms; namely, 
 cohesion, elasticity, and perhaps others, with chemical 
 energy as a precursor or concomitant. The exercise of 
 muscular energy is in general controlled by the brain and 
 other nerve centers. But here again there appears to 
 be no introduction of energy of other than the foregoino- 
 
Chap. IV.] FORMS OF ENERGY. 37 
 
 forms. The term vital energy is also in somewliat general 
 use in a non-scientific way, to denote the energy of living 
 things. There appears to be no evidence that there is here 
 another form. Eather, the term seems a convenient one to 
 include the sundry known forms which, in association, pro- 
 duce the phenomena characteristic of animate things. A 
 condition certainly exists in animate objects, in virtue of 
 which energy-changes occur which would not occur other- 
 wise. But to what this condition is due, remains to be 
 discovered. The experimental difficulties, attendant upon 
 investigations of energy-changes in living organisms, have 
 prevented progress in that direction, so that perhaps the 
 most that can at present be asserted is that there is no 
 evidence of any form of energy peculiar to living things. 
 Hence muscular and vital energy are not terms denoting 
 independent forms of energy. 
 
 List of Forms of Energy. — We have thus demonstrated 
 the existence of at least nine forms of energy which, so far 
 as we have yet proved, are distinct from one another ; these 
 are: — 
 
 1. 
 
 Kinetic Energy, 
 
 6. 
 
 Chemical Energy, 
 
 2. 
 
 Gravitation Energy, 
 
 7. 
 
 Electrical Energy, 
 
 3. 
 
 Heat, 
 
 8. 
 
 Magnetic Energy, 
 
 4. 
 
 Energy of Elasticity, 
 
 9. 
 
 Eadiant Energy. 
 
 5. 
 
 Cohesioa Energy, 
 
 
 
 This list includes all known separate forms. 
 
 To determine how far these nine forms are mutually 
 independent, demands a knowledge of the nature of the 
 various forms. This knowledge, science does not now fully 
 jjossess. Theories there are, but of direct knowledge, next 
 to nothing. These theories indicate that heat is molecular 
 kinetic energy; that cohesion and gravitation energy are 
 identical ; that radiant energy is a form of vibration energy 
 (of a substance called the ether) ; but these theories are not 
 
38 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 a sufficient warrant for simplifying the classification. Hy- 
 pothesis goes even further and suggests that each form 
 arises from a mode of motion of some portion or portions of 
 substance or of matter, and that therefore all energy is 
 kinetic. This proposition will be recurred to, but however 
 alluring it may seem, the fact that it rests only on hypoth- 
 esis must not for an instant be overlooked. It is often 
 accepted as a personal belief, but beliefs frequently rest on 
 even more slender foundations than does this one, and are 
 not to be propounded as scientific facts. 
 
 No assumption will be here employed to the effect that 
 all energy either is or is not kinetic in its nature. It may, 
 however, be said that in the present state of our knowledge, 
 while it would be distinctly unsafe to assert that all energy 
 is kinetic, it would not be unsafe to assume as a working 
 hypothesis that it may be. To many minds, such a hypoth- 
 esis would remove a sense of mystery or vagueness, and to 
 most persons perhaps it not only would prove a decided 
 help, but, if always held in leash as a hypothesis, it would 
 lead into no serious misapprehension. 
 
 Classification of Energy. — The above classification of forms 
 of energy is based upon the properties which substances 
 exhibit. This seems to be a natural mode of classification. 
 A decidedly different method has been recently advocated 
 by some German writers. It may be here remarked, how- 
 ever, that as the term form has been so long appropri- 
 ated in the ordinary classification based on properties of 
 bodies, it would seem to have been wiser to select another 
 term by which to designate the classes under this new 
 system. Mention may here be made of the division of all 
 energy into two groups, kinetic and potential. This will be 
 considered in another section. 
 
 Mechanical Energy. — Those forms of energy which play 
 a part in the operation of machinery are roughly grouped 
 in a class called mechanical energy. This is not a definite 
 
Chap. IV.] FORMS OF ENERGY. 39 
 
 or important classification, but one which is more or less 
 familiar. Perhaps it may properly be said to include 
 other forms than heat, chemical, electrical, and magnetic 
 energy, that is, any other than these four forms might be 
 referred to as mechanical energy, although radiant energy 
 would hardly be so classified. 
 
CHAPTEE V. 
 
 FORCE. 
 
 Definition of Force. — The state of motion of a body at a 
 given time is fully determined by the velocity and direction 
 of the motion at the given instant. The motion is to be 
 referred to any point not involved in the change of motion 
 to be observed. This is the most general mode of reference 
 possible, and therefore the most convenient. The definition 
 of energy (p. 20) states without qualification that the power 
 to change the state of motion of a body is energy, and it 
 therefore follows that nothing other than energy can produce 
 that change. 
 
 Energy may exist without producing, or even without 
 tending to produce change of motion ; e.g. the kinetic energy 
 of a freely moving body. When, however, change of motion 
 is taking place, energy in some form must be acting upon 
 the body to produce that change. Energy may be in action 
 against a body without changing the motion, owing to some 
 restraint upon the body's freedom of motion. In that case 
 the energy will be producing merely a tendency to change 
 of motion, by which we mean that change will be produced 
 if the restraint is removed. This tendency to change may 
 exist not only when the body is at rest but also when it is in 
 motion of any sort. Thus energy may be producing a ten- 
 dency to change of motion in a body, or portion of substance, 
 whether free, or constrained, and in any state of motion 
 whatever; rest being a special case of motion. This action 
 
 40 
 
Chap. V.] FORCE. 41 
 
 of energy is force, and force can be produced by nothing but 
 energy. We may, therefore, define force thus : — 
 
 FOECB IS THAT AcTION OF EnEKGY BT WHICH IT PKODUCBS 
 
 Tendency to Change in State of Motion of Bodies. 
 
 The relation between this definition and that of Newton 
 and others will be considered in Chapter XII. 
 
 The relation and distinction between force and energy ex- 
 pressed in this definition cannot be too strongly emphasized. 
 Energy is the agent, — the doer, — the cause, — that which 
 performs the action and produces the effect. The effect 
 produced is " tendency to change in state of motion," which 
 is a condition thus imposed on the body. The operation, 
 process, effort, mode of acting, action, by which energy pro- 
 duces this effect is force, as here defined. Force is thus that 
 action by which the cause — energy — produces a particular 
 effect ; namely, tendency to change of state of motion. And 
 action is used in its generally recognized sense of operation, 
 etc., as just stated, not in the sense of effect. 
 
 The operation of imposing upon or maintaining against a 
 body the condition denoted by " tendency to change in state 
 of motion," in other words, the exertion of force, demands 
 no expenditure of energy by the acting source of energy. 
 For example, a compressed spring expends none of its 
 energy of elasticity, however long it continues to exert 
 force (pressure) against the objects which are preventing 
 its recoil. The moment, however, that any portion of the 
 restraint is removed, an action of the elastic energy occurs 
 which is not force but which does involve the expenditure 
 of some of that energy, — a process to be described later 
 imder the name of worlc. To fix still more clearly the limit 
 of the meaning of force, we may anticipate slightly. When- 
 ever a given source of energy is producing upon a given 
 body tendency to change in state of motion, it is by defini- 
 tion exerting force against the body, whether or not the 
 
42 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 body is in motion of any sort whatever. The mere exertion 
 of the force withdraws none of the energy ; and the force, 
 being the action, continues only so long as the action en- 
 dures. If the body has partial or complete freedom of 
 motion in the direction of the force (the direction in which 
 the body would move if entirely free), then the action of the 
 energy will be twofold; the energy will both exert force 
 and "perform work." The latter process consists in the 
 transference of energy from the acting source to the body 
 acted upon, as will be duly shown. Both force and work are 
 thus names of operations or processes, and do not denote 
 forms or effects of energy. Neither of them is an entity, 
 or quantity having permanent existence, in the sense in 
 which energy and matter are often called entities. Force 
 can produce nothing in the sense in which a cause is said 
 to produce an effect. Energy is the cause which, through 
 the action called force, produces the effects so often errone- 
 ously ascribed to force ; namely, the various changes in state 
 of motion of bodies. 
 
 It is a most important result of general observation that, 
 when energy is exerting force, two or more bodies are always 
 involved in the action. That is, wherever one object is 
 observed to possess a tendency to acceleration in a certain 
 direction, another object is always discoverable (directly or 
 indirectly) towards which or away from which the first 
 tends to move, and which also has a tendency to motion 
 towards or away from the first object. Often more than 
 two objects are involved in the same action upon another. 
 Familiar examples are as follows. In the collision of two 
 elastic balls, the elastic energy of each exerts an equal force 
 upon the other. Gravitation energy exerts a force tending 
 to move the planet towards the sun, but also exerts an equal 
 and opposite force tending to make the sun move towards 
 the planet. A body near the earth's surface is urged by 
 gravitation energy towards the earth, but the earth is urged 
 
Chap. V.] FORCE. 43 
 
 by gravitation energy with an equal force towards the body. 
 Wherever change of state of motion of one body with ref- 
 erence to another is in progress, force is being exerted by 
 energy upon one or the other of the two bodies, but not 
 necessarily upon both. From the consideration of the mere 
 motion of these two bodies, it is not possible to determine 
 upon which the energy is exerting the force. But by scru- 
 tiny of the surroundings or by extended investigation, it is 
 generally possible to arrive at a correct decision on this 
 point. It is, however, evident that in endeavoring to ascer- 
 tain the precise locality and amount of the real forces in- 
 volved in any action, a choice of reference points for the 
 motion is of the utmost importance. Mere acceleration 
 affords knowledge and measure in many cases of only a 
 portion of the force, either by reason of the choice of ref- 
 erence point or by reason of constraint in the motion of 
 the bodies. 
 
 How Energy produces Force. — The actual manner in which 
 energy produces the action which we call force must ob- 
 viously depend upon the nature of energy itself, and of the 
 particular form of energy exerting the force. Upon this 
 point very little can yet be discovered through direct ob- 
 servation. We know that kinetic energy is transferred and 
 force exerted in the collision of bodies. But we also know 
 that the transference is through the energy of elasticity, 
 and that it is elastic energy which exerts the force between 
 the two bodies. Of the intimate nature, or mechanism, of 
 elastic energy nothing is directly known. We have only 
 hypothesis to assist us here ; and the Vortex-ato7n Hypothesis 
 described later is the only one which affords an ultimate 
 explanation of elasticity by accounting for it through a 
 peculiar mode of motion. The various atomic hypotheses 
 explain the elasticity of bodies by assuming perfectly 
 elastic molecules, out of which the bodies are supposed to 
 be constructed. They fail to account for the elasticity of 
 
44 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 the molecules. An examination of the statement of these 
 theories given in Part II. will show how they account for 
 certain forces as consisting of a bombardment by elastic 
 molecules. This idea is a helpful one, but it must always 
 be remembered that it is merely a hypothesis. The student 
 of physics cannot be too caref id in discriminating between 
 such hypotheses as the molecular theories or deductions 
 from them, and such fundamental facts as the laws of 
 energy and of force as an action of energy. 
 
 Recognition of Forces. — We become conscious of the action 
 of some forms of energy through our senses. Thus we are 
 conscious of the force called weight when we hold a heavy 
 object in the hand ; and in various ways we recognize press- 
 ures which are exerted upon us by external objects. This 
 consciousness does not necessarily imply what may be 
 termed a sense of force ; all that is here meant is that we 
 are conscious of a muscular effort which is necessary to 
 resist an external force, or to exert a corresponding force 
 u.pon surrounding objects. We can gain through sensation 
 a rough measure of the amount or quantity of force exerted 
 upon or by ourselves. The purely physical methods, how- 
 ever, for the recognition of force are two. First, by the 
 change in state of motion of bodies ; second, by the distor- 
 tion of bodies. 
 
 From our definition of force, it is obvious that when we 
 can see a change in the state of motion of one body with 
 reference to another, we know that force is being exerted 
 upon one or both of them. If, however, either or both of 
 the bodies are under recognized restraint relatively to each 
 other, and we find that on removing that restraint a change 
 of motion occurs whenever and as often as the restraint is 
 removed, then we infer that force is being exerted upon one 
 or both during the time of restraint. 
 
 If a spring be bent, or if any elastic body be strained out 
 of shape, it tends to regain its former unrestrained shape or 
 
Chap. V.] FORCE. 45 
 
 position. If when so strained it be set free, it will return 
 to its normal form unless it has been strained beyond the 
 limits of perfect elasticity. In so doing it will change the 
 state of motion of its own parts and of other bodies con- 
 nected with it. It is, therefore, exerting force when under 
 strain. This is universally true of all solid bodies ; so that 
 whenever we observe deformation to exist in any solid, we 
 know that the body must be exerting elastic force. This fur- 
 nishes us with our second physical means of detecting force. 
 
 Resistance. — Whenever a body is not free to move in 
 respect to any energy acting upon it, we are accustomed to 
 say that there is some " resistauce " opposing its freedom. 
 If we inquire into the nature of this resistance, we shall see 
 that it is always due to the action of some form of energy, 
 the action being of such a nature as to constitute a force 
 opposing the body's motion. 
 
 Tlie sufficient evidence that all resistance is due to the action 
 of energy lies in the fact that through resistance change in state 
 of motion of bodies occurs. 
 
 That which we name resistance, however, is not the 
 energy, but merely that action of it which produces the 
 tendency to change in state of motion. Therefore, by our 
 definitions : — 
 
 All resistance is force. 
 
 In raising a body from the earth, we encounter a resis- 
 tance. This we know as weight, a force due to the energy of 
 gravitation. In moving a piece of iron away from a mag- 
 net, we encounter a resistance ; here too we know that re- 
 sistance is a force, — magnetic force due to magnetic energy. 
 In both these instances the resulting force exists indepen- 
 dently of the motion against it ; but it often occurs, as in all 
 frictional resistance, that the force is called into play by the 
 motion which it opposes. Eesistances which are plainly 
 due to well-recognized forces, such as weight, electrical and 
 magnetic forces, and so on, we do not need to consider 
 
46 FUNDAMENTAL PBTSICAL CONCEPTS. [Pakt I. 
 
 further; but it may be •well to inquire into the nature of 
 the resistance of friction. We shall find that this appears 
 to be due to an opposing force arising from the elasticity of 
 the materials involved, and ultimately results in the accel- 
 eration of particles of bodies, producing heat energy. 
 
 External Friction of Solids. — The friction of rubbing solid 
 surfaces is due to their roughness. When solid surfaces, 
 even if smooth, are examined under high magnifying power, 
 they will be seen to be covered with a multitude of scratches 
 and projecting points. In rubbing surfaces these irregulari- 
 ties interlock. As the one surface is drawn over the other, 
 the projecting parts bend each other backward, bringing 
 into play the force of elasticity. As the motion progresses, 
 these parts bend sufficiently to slip past one another, or the 
 bodies are more or less forced apart. Thus the resistance 
 of solid friction is due in part to the force produced by elas- 
 tic energy locally called into action at the points of contact. 
 The energy expended in maintaining motion against friction 
 is thus converted directly into energy of elasticity ; then as 
 these interlocking points slip past one another, they enter 
 into vibration, the energy of which is soon transformed in 
 turn into heat. This transformation takes place, in large 
 part, within the substance itself through its imperfect elas- 
 ticity, that is, through the presence of internal friction, the 
 nature of which will be presently described. Another por- 
 tion of the vibration energy is transmitted (sound waves) to 
 surrounding bodies and is there similarly transformed into 
 heat. 
 
 In the rubbing of two solid surfaces, the interlocking por- 
 tions are more or less broken and scratched. These pro- 
 cesses consist essentially of the separation of portions of the 
 solid against the action of cohesion energy, as will be pres- 
 ently shown, and the energy thus expended remains in part 
 as cohesion energy, and is partly reduced through vibration 
 and imperfect elasticity to heat. 
 
Chap. V.] FORCE. 47 
 
 Two rubbing surfaces are ordinarily held in contact by- 
 weight or some external force ; but when the surfaces ap- 
 proximate to perfect planes, or at any points where the 
 contact between the two is very close, the force of cohesion 
 undoubtedly comes into play by holding the surfaces still 
 more firmly together, thus increasing the friction. At such 
 points of intimate contact, internal friction undoubtedly 
 plays a part. 
 
 Internal Friction of Solids. — A hypothetical explanation of 
 this phenomena is as follows : The molecules of a solid are 
 in rapid but restricted vibration irregularly in all directions. 
 The kinetic energy of this vibratory motion constitutes the 
 heat present in the body. Imagine a geometrical plane to 
 be passed through the solid body. Particles from either 
 side of the plane are continually passing through to the other 
 side, coming into collision with particles of that side, and 
 returning usually to their original place. Suppose now one 
 part of the solid is being drawn over the other on this 
 intersecting plane. For convenience suppose the plane to 
 be horizontal. Then particles from the upper half will be 
 continually passing into the lower half, carrying with them, 
 on the average, the onward velocity of the upper half. On 
 the average, these particles through their contact with those 
 in the lower half will give up the energy corresponding to 
 this onward velocity, and will return to the upper portion 
 without it, and while there again take it up. There is thus 
 a continuous passage of energy from the moving portion to 
 the under portion which we will consider as stationary, that 
 is, as the point of reference for the motion. This transferred 
 energy will, in the lower half, assume the form of heat, 
 since it will result in the increase of the kinetic energy of 
 the particles. Although the velocity from which this energy 
 arises is wholly in one direction, it will soon become irregu- 
 larly distributed throughout the mass, as this is fixed in 
 position. The energy thus becomes irregularly distributed 
 
48 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 kinetic energy of the particles, and thus is of the form of 
 heat. 
 
 Equally the particles below the plane are passing through 
 to the upper side, and returning with an increase of energy 
 acquired in -virtue of the onward motion of the upper por- 
 tion, and at the expense of the energy maintaining the mo- 
 tion; this added energy is at once converted into heat 
 through collisions, and thus more of the acting energy is 
 transformed. 
 
 AYhen the great number of the interchanging particles per 
 unit area, which the kinetic theory of heat requires, is con- 
 sidered, it is easy to understand how a very large expendi- 
 ture is necessary to maintain such motion in a solid. In 
 brittle solids, this motion can be carried to only a very 
 slight extent, owing to the occurrence of fracture. In im- 
 perfect and ductile solids, and in general in all solids as 
 they approach their melting-point, internal motion can be 
 carried further without rupture, and the internal frictional 
 resistance diminishes. 
 
 Since the energy maintaining the motion is expended in 
 continuously accelerating the motion of the particles in the 
 direction of motion, it must be exerting a force in that di- 
 rection ; also since the acceleration is imparted to the parti- 
 cles through their elasticity, there is a force of elasticity 
 tending to retard the upper and to accelerate the lower 
 (fixed) portion. This action constitutes a part of the inter- 
 nal friction of solids. Another part of the internal friction 
 of solids — or more properly another way in which energy 
 is expended in maintaining the motion referred to — arises 
 through cohesion. Every particle of a solid is under the 
 action of cohesion energy, which tends more or less strongly 
 to make it approach each of its neighbors. On the average, 
 the total force is the same in every direction upon any one 
 particle, so that the resulting force is zero ; but when mo- 
 tion of one portion of a solid takes place over an adjacent 
 
Chap. V.] FOTtCE. 49 
 
 portion, as in the above example, each particle above the 
 separating plane is torn from its association with the parti- 
 cles behind it below that plane, and is brought into equally 
 intimate association with those particles to which it is 
 approaching and which also lie below the plane. So long 
 as the average distance of the particle from its neighbor 
 remains constant, the mere change of position requires no 
 resulting expenditure of external energy ; since the energy 
 necessary to separate the particles against cohesion is ex- 
 actly equal to that restored by cohesion as the particles ap- 
 proach others ; but the recession from one particle and the 
 approach to another, under a force which varies inversely 
 as some function of the mutual distance, will in general set 
 up in the particles, irregularly directed vibrations whose 
 energy will add to the other heat energy of the body. Thus 
 further demand will be made upon the energy-source which 
 is maintaining the relative motion of the two parts of the 
 body. This constitutes the remainder of the internal fric- 
 tion of solids. 
 
 If the solid is torn apart at such an imaginary plane; 
 that is, if the two portions are separated so far that the 
 force of cohesion is no longer sensible, then a part of the 
 energy so expended is stored up as cohesion energy, to be 
 restored if the parts are at any time brought near enough 
 together for it to act. Another part is transformed into 
 energy of vibration and thence into heat as just described. 
 
 Internal Friction of Fluids. — In liquids and gases the 
 nature of the internal friction is substantially that of solids, 
 except that the cohesion is nearly or quite insensible in its 
 influence. There is also the difference that the particles do 
 not oscillate about a mean fixed position, but wander with 
 more or less rapidity and freedom throughout the volume 
 of the substance. This, however, makes no important dif- 
 ference in the nature of the phenomenon. 
 
 Gases and liquids oppose a still further resistance to the 
 
50 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 motion of a solid body through them, by reason of their dis- 
 placement by the moving body. To set the displaced mass 
 of fluid in motion requires an expenditure of energy. The 
 kinetic energy thus imparted is rapidly reduced from a 
 more or less irregular molar form to the irregular molecular 
 form of heat, through the internal friction of the fluid. 
 Also there is the hydrostatic pressure to be considered. 
 When the submerged body is at rest relatively to the fluid, 
 the pressure is equal in all directions at equal depths, if due 
 to weight, but if the body is in motion, the pressure upon it 
 in the direction of motion is lessened in proportion as the 
 velocity is greater. Tor the energy causing the pressure 
 acts only at a definite space-rate and can, therefore, acceler- 
 ate the fluid as it follows up the moving solid at only a defi- 
 nite maximum rate. Part of the primary energy being thus 
 expended in accelerating the motion of the fluid, less re- 
 mains to exert force upon the moving solid ; and when the 
 acceleration of the fluid absorbs the primary energy at its 
 full space-rate, then the hydrostatic pressure in the direc- 
 tion of the motion of the body ceases entirely. Thus the 
 resisting hydrostatic pressure upon a solid moving in the 
 fluid is zero when the motion is nil, and increases to a maxi- 
 mum when the motion is sufficiently rapid to bring about 
 the condition above mentioned. For higher velocities the 
 pressure remains constant. 
 
 There is also, in the motion of a solid through fluids, the 
 surface friction, which is of the same general character as. 
 the surface resistance between solids. 
 
 Stress, Attraction, Repulsion. — "When we consider the 
 action of energy in producing tendency to acceleration with 
 reference merely to the body acted upon, we ordinarily 
 employ the term force to represent the action ; when, how- 
 ever, the energy is acting to cause two bodies to approach or 
 recede from one another and we are considering the action 
 upon both of two bodies, we call it a stress, that is, we say 
 
Chap. V.] FORCE. 51 
 
 there is a stress between the two bodies ; or if the energy- 
 is acting upon two parts of tlie same body, we speak of a 
 stress in the body. If the action tends to move the body or 
 particle away from the supposed location of the energy, we 
 call the force a pressure ; if the action is regarded as tend- 
 ing to move the body or particle towards the supposed loca- 
 tion of the energy, we call the force a tension. When we 
 regard the energy as tending to move bodies or particles fur- 
 ther apart, the stress is called repulsion or sometimes press- 
 ure. When the stress tends to make two bodies or particles 
 approach one another, it is called attraction or sometimes 
 tension. A pressure may be called a push and a tension a 
 pull. We speak also of a force of attraction when we are 
 considering only one portion of the attractive stress, that is, 
 the action upon one of the bodies or particles concerned ; 
 and similarly we speak of a force of repulsion. When elas- 
 ticity is in operation, if there is compression, the stress is 
 composed of equal pressures in opposite directions upon the 
 two bodies or particles involved ; or, if there is extension, 
 the stress consists of equal attractions. It may be added 
 that most forces appear to be of the nature of elastic force 
 or stress, and it is generally conceded that force can be 
 conceived of only as a pressure. 
 
 When we consider the action of gravitation upon bodies, 
 we find no direct evidence of any stress in which the bodies 
 take part ; that is, when the gravitation energy is accelerat- 
 ing a body, it is exerting force upon that body, but there is 
 no evidence that the body is exerting a force against gravi- 
 tation ; in fact we distinctly conceive it not to be doing so. 
 Yet no hypothesis of gravitation has been framed which 
 accounts for the gravitation of bodies without ascribing to 
 the particles of the gravitating bodies, and also to the sub- 
 stance causing the gravitation, the property of elasticity. 
 This fact points not only to the possibility suggested by 
 other considerations, that matter does not inherently possess 
 
52 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 the property of gravity, but that this property attaches only 
 to portions of matter which have that energy associated 
 with them which is necessary to constitute them bodies 
 (molecules) as distinct from mere matter (cf. Chaps. VII. 
 and XI.). 
 
 The use of the phrases, "a body attracts another," or "a 
 body repels another " and similar ones, is somewhat mislead- 
 ing, since the phrases imply that the cause of the attraction 
 or repulsion is resident upon the bodies, whereas in general 
 we do not know the locality of the energy. They cannot, 
 however, be excluded from use, and it therefore becomes 
 necessary to remember that the foregoing implication may 
 be contrary to fact. 
 
 Force as an Index of Energy. — Force being the action of 
 energy, its presence is of course sufficient evidence, at any 
 time, of the existence of a form of energy causing it. "We 
 need, therefore, no further proof of the presence of energy 
 than the occurrence of some phenomenon which we know to 
 accompany the presence of force. It is not, however, to be 
 forgotten that our fundamental proof of the existence of 
 force goes back to the test as to whether acceleration would 
 occur under the given condition in a free body. 
 
 Remarks upon Force. — It is of the utmost importance for 
 clear thinking to fix in the mind the fact which the pro- 
 posed definition of force (p. 41) brings out; namely, 
 that force is only a manifestation of energy. It is essen- 
 tially secondary in its nature. Force can have no existence 
 independently of energy. It cannot produce energy or 
 destroy it. It is not a thing which can be transformed or 
 which has any existence in the sense in which matter and 
 energy exist, as indicated through the p)rinciples of the con- 
 servation of matter and energy. For brevity, it has been 
 customary to speak of forces and bodies as doing certain 
 things or performing certain actions. But it must be re- 
 membered that the thing which does or performs is really 
 
Chap. V.] FORCE. 53 
 
 the energy and not the force or body. It is the energy 
 exerting the actions called force and work. 
 
 The utmost confusion prevails, in physical works of all 
 kinds and still more in ordinary language, in the use of the 
 terms force and energy. Unhappily our text-books of 
 physics do little to remedy this state of affairs, and more 
 often add to the confusion than relieve it. This is a natural 
 inheritance from the state of physical knowledge of a half 
 century or more ago. At that period various "Physical 
 Forces," so called, were clearly recognized. These were. 
 Motion, Heat, Electricity, Light, Magnetism, Chemical 
 Affinity, and "Other Modes of Force" [Grove]. The idea 
 of energy had not then been adopted, and what we now call 
 the energy of a moving body, that is, its kinetic energy, was 
 called " the force of motion " of the body. When such a 
 moving body struck a resisting object, "the force of the 
 blow " was the expression used to refer rather to the energy 
 of the striking body than to the force proper which it ex- 
 erted while it was being checked in its motion by the resist- 
 ance of the object. Thus even after the distinction between 
 force and energy began to be recognized, and the principle 
 of convertibility to be apprehended {e.g. the transformation 
 of heat into force of motion, kinetic energy, or vice versa), 
 there still remained a great degree of confusion. This 
 arose both from a lack of acquaintance with energy and 
 force, and from the want of a clear definition of either. 
 The confusion at the present time results almost wholly 
 from the latter cause ; for with a correct definition of force, 
 it entirely vanishes. 
 
 The comprehension of physical phenomena becomes rela- 
 tively simple and satisfactory if we accustom ourselves to 
 thinking : first, of matter as absolutely inert, and therefore 
 as incapable, of itself, of exerting force or any action what- 
 ever; second, of energy as a power conferred upon matter 
 either by the bestowal of motion to it (cf. Part II.) or in 
 
54 FUNDAMENTAL PBYSICAL CONCEPTS. [Paet I. 
 
 some unknown way, and as that to which all changes in the 
 appearance or properties of inanimate things are due ; and 
 third, of force as an action of energy. 
 
 Since force is an action of energy and not something 
 which in itself possesses a power to change the state of 
 motion of bodies, confusion in the minds of beginners might 
 be avoided if the term force were to be employed in the 
 passive rather than in the active sense. Thus, we should 
 speak of force as being exerted upon or against an object 
 rather than as acting upon it ; of motion, as arising from 
 the existence, or presence, or application of the force rather 
 than from the action of a force. This minor point is well 
 worth the attention of those who are teaching the elements 
 of the subject, and a little thought will show that no 
 more circumlocution is necessary in employing the passive 
 than the active form of expression, although it is by no 
 means easy for one who has become accustomed to the 
 latter to change to the former. To one who has at all mas- 
 tered the subject, the phraseology is comparatively, though 
 not entirely, unimportant. The difficulties of acquiring a 
 clear conception of physical ideas are not so small that 
 teachers can afford to overlook this point, even though at 
 first sight so insigniiicant ; indeed an imperfect understand- 
 ing of the term force and of its relation to energy is perhaps 
 the greatest stumbling-block in elementary physics. 
 
CHAPTER VI. 
 KINETIC ENERGY, KINERGETY. 
 
 Apparatus. — The law of kinetic energy may be illustrated, 
 as has been stated, by the collision of elastic balls. The 
 method however is not general, since it deals only with 
 bodies of nearly perfect elasticity. Also it is hardly suffi- 
 ciently simple for elementary demonstration. Many other 
 methods might be suggested, but the following has some 
 advantages. 
 
 The apparatus consists chiefly of a spring buffer so de- 
 signed that it can take up substantially all of the kinetic 
 energy of an object impinging perpendicularly upon its 
 surface. This energy is stored up as energy of elasticity in 
 the spring of the buffer, and can be subsequently imparted 
 in the form of kinetic energy to any free body resting 
 against the buffer by merely releasing the spring. One 
 form of the buffer may be briefly described thus. Two up- 
 right metal plates ten or twelve inches apart and one or 
 two inches square are fastened to a base. Through guiding 
 holes in these plates passes a straight steel rod or hard- 
 brass tube about eighteen inches long and as light as is 
 consistent with stiffness. The forward end of this rod car- 
 ries a round, flat, circular plate an inch and a half in diam- 
 eter, preferably of steel. This head-plate projects about 
 four inches from the forward guide. Fastened to the rod 
 just behind the forward guide is a collar, and extending 
 from the collar to the rear guide and closely surrounding 
 the rod is a spiral spring of well-tempered steel. Behind 
 the rear guide is some sort of a clutch which shall catch and 
 
 55 
 
56 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 hold the rod as it passes through the rear guide by any push 
 upon the head-plate. The spring tends to force the head- 
 plate and rod forward until the collar strikes the front 
 guide, and to hold it there. The clutch must act so 
 promptly and effectively as to hold the rod at sensibly the 
 furthest point which it reaches, and must be so arranged 
 that it can be completely released when desired, giving the 
 rod entire freedom to move forward until stopped by the 
 collar. One form of clutch which is suflB.ciently effective 
 consists of a taper wedge which shall slide point downward, 
 back of the rear guide, by its own weight, the slope of the 
 wedge bearing always against a pin projecting at right 
 angles to the rod. A ring-clutch such as is used on many 
 arc lamps is better, but rather more difficult to construct. 
 The spring must be of such stiffness as to be compressed 
 tliree or four inches by the most powerful blow to which 
 the buffer is to be exposed. 
 
 The objects whose kinetic energy is to be experimented 
 with are best made in the form of balls, and are suspended 
 by a cord of any desired length, or better by two cords 
 diverging upward. The point of attachment of these 
 ■ cords at their upper ends should be such as to allow the 
 body to rest lightly against the head-plate of the buffer 
 when the clutch is released and the body hanging free. If 
 such a suspended body be dra^v^l backward in a vertical 
 plane through the rod of the buffer and allowed to swing 
 free, it will strike the face of the buffer and be brought to 
 rest by the spring. Its kinetic energy relative to the buffer 
 will have been entirely given up. This energy will have 
 been mainly transformed into energy of elasticity in the 
 spring, and this will be held stored up there through the 
 action of the clutch which prevents the spring from giving 
 it up. Small portions of the energy will have been other- 
 wise expended. Some will have been transformed into 
 heat through friction of the moving parts; some into sound 
 
Chap. VI.] KINETIC ENERGY, KINERGETY. 57 
 
 vibration ; some into cohesion energy and heat in the strik- 
 ing body as well as in the buffer, owing to imperfect elas- 
 ticity ; and finally, some into energy of elasticity in the 
 body itself. The last-named portion will by its action 
 against the face of the buffer push the body away, causing 
 a slight rebound. The energy of this rebound must be de- 
 ducted, in careful quantitative experiments, but will ordi- 
 narily be so small as to be negligible in class-room work. 
 The same remark holds with reference to the other por- 
 tions of energy which are lost. Of course if the final 
 position of rest of the buffer after the blow is not such 
 that the object when in contact with it is directly below its 
 point of suspension, the object will swing back as a pen- 
 dulum. In more careful work, therefore, it is necessary to 
 adjust suitably the position of the buffer to avoid this 
 slight disturbance. 
 
 After such a blow any other object may be substituted 
 for the first, and the clutch then released. The elastic 
 energy of the spring will then impart kinetic energy to the 
 object, slight losses occurring in the transfer similar to 
 those above named. To show approximately what the loss 
 of energy is in the process, a given object moving at a 
 definite velocity may be allowed to strike the buffer, and 
 the energy thus transferred to the buffer may be afterwards 
 restored to the same object. The object should leave with 
 the same velocity as that with which it struck. The dif- 
 ference in its velocity, or rather the corresponding differ- 
 ence in the amount of its energy, the computation of which 
 will be shown later, gives the loss in the operation. The 
 amount of this loss, however, varies with the amount of 
 energy employed, and the nature of the substance of the 
 striking object. 
 
 The relative velocities on striking and upon leaving the 
 buffer can be determined by the ordinary experimental law 
 of the pendulum. The square of the velocity at the lowest 
 
58 FUNDAMENTAL PHYSICAL CONCEPTS. [Part 1. 
 
 point of its swing is proportional to the vertical distance 
 through -which the object descends. Also, if the suspension 
 is several times the length of the arc through which the 
 object swings before or after striking, the vertical distance 
 through which it descends is nearly proportional to the hori- 
 zontal distance through which it swings. 
 
 The foregoing apparatus, even without the application of 
 corrections for minor losses, will serve sufficiently well for 
 the following illustrations. It will of course give a much 
 closer approximation with the use of the corrections. It 
 is sufficiently exact to justify the assumption made in the 
 following demonstrations that (sensibly) all the kinetic 
 energy of one body may be expended in producing kinetic 
 energy in another body. The transference is, to be sure, 
 not a direct process, but this in no way concerns the cor- 
 rectness of the proposition ; moreover, in fact, as may be 
 seen by an examination of all imaginable cases of the trans- 
 ference of kinetic energy, direct transference never occurs. 
 
 Quantity of Kinetic Energy. — In defining quantity of 
 kinetic energy we must proceed directly from the funda- 
 mental definition. Since that definition bases the recogni- 
 tion of energy upon change in the state of motion of 
 bodies, quantity of energy must be primarily defined 
 through amount of change in state of motion. As all 
 change in state of motion may be resolved, as shown in the 
 chapter on Motion, into acceleration from a state of rest, 
 such acceleration is naturally adopted as the basis of 
 primary measurement. Also as free motion is the simplest 
 case of motion, the fundamental measurement must be 
 through the acceleration of free motion of the body. An 
 axiom which must lie at the basis of the measurement is 
 then : — 
 
 To accelerate equally from rest the free motion of a con- 
 stant body at different times, equal quantities of energy must 
 be imparted to the body. 
 
Chap. VI.] KINETIC ENERGY, EINERGETY. 59 
 
 This axiom affords a test for, or defines, equal quantities 
 of kinetic energy. 
 
 By a constant body or the same body will be meant a 
 body wbicli is undergoing no discoverable change in any of 
 its properties during the time of experiment. Pieces of 
 metal, quartz, glass, etc., are common examples of sensibly 
 constant bodies. By a standard body is meant some con- 
 stant body selected and preserved for reference. Such 
 standard bodies are the International Kilogramme and the 
 Imperial Pound, represented in practice by their metallic 
 copies. 
 
 The first proposition to be proved in studying the law of 
 kinetic energy is this : — 
 
 The same body moving with equal velocity at different times 
 possesses the same quantity of kinetic energy. 
 
 This may be illustrated by the apparatus described in a 
 preceding paragraph. Let a constant body A give up all 
 its kinetic energy to the spring buffer ; and let this energy 
 be subsequently given out in accelerating from rest a sec- 
 ond free body B. In successive trials, the resulting velocity 
 of B will be found the same whenever the velocity of A on 
 impact is the same. This observation is confirmed by gen- 
 eral experience. 
 
 Capacity for Kinetic Energy: Kinergety.^ — The second 
 proposition to be established is : — 
 
 Any given body possesses a definite capacity for kinetic 
 
 1 The importance of enforcing the concept of capacity or receptivity for 
 kinetic energy, and the awkwardness of the phrase itself, justify the intro- 
 duction of a new term. The word kinergety seems well adapted to the 
 purpose. It will be used both in the abstract sense, to denote the idea 
 of capacity for kinetic energy, and in the concrete sense, to denote the 
 quantity of this capacity; following in this respect the analogy of energy 
 and many other terms. The etymology of the word is sufficiently 
 obvious. 
 
 Certain other terms similarly derived, and perhaps also new, may be 
 suggested as being of possible service. These are : — 
 
 Energia = energy, or, perhaps better, quantity of energy. Such a term 
 
60 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 energy, that is, a definite Idnergety ; and this is constant in 
 amount so long as the body is constant. 
 
 The term kinergety will be employed to denote capacity 
 for kinetic energy (cf. Chap. XI.). By the possession of a 
 definite kinergety is meant, that a body moving at different 
 times with the same velocity will possess the same amount, 
 neither more nor less, of kinetic energy. In the case of 
 many constant bodies this proposition is capable of direct 
 proof or illustration by the apparatus described in the 
 former paragraph. In all cases it may be inferred from 
 observed phenomena. 
 
 Different bodies in general possess different kinergeties. 
 
 This third proposition is obvious in cases where we build 
 up bodies from smaller parts. It may be in general proved 
 in the same way as the preceding proposition. 
 
 This fact that each individual object or body (that is, 
 every portion of every substance) possesses an individual 
 and fixed kinergety, is one of most fundamental impor- 
 tance. The kinergety is or appears to be a permanent prop- 
 erty of substance, and appears to inhere in each individual 
 portion quite independently of the existence of any other 
 portion. There is no evidence that its presence is due to 
 anj-thing external to the substance or to anything associated 
 with it. On the other hand, so far as we can discern, to 
 annihilate this capacity would be to annihilate the sub- 
 stance simultaneously. But however natural this inference 
 
 would be very serviceable in physios, and its adoption would be in line 
 with recent convention with regard to such terms as induction and induc- 
 tance, etc. 
 
 Energies possessing energy, — an adjective form now obsolete but well 
 adapted for use in physics where energetic would be unsuitable because of 
 its significauce of capacity or ability to exert energy. 
 
 Einergy and kinergia, to denote respectively kinetic energy and quantity 
 of kinetic energy. 
 
 Kinergic = possessing kinetic energy. 
 
 Klnerg = a unit quantity of kinetic energy; e.g. one-half of the kiner- 
 gia of one gramme moving with a velocity of one centimeter per second. 
 
Chap. VI.] KINETIC ENERGY, KINEBGETY. 61 
 
 may seem, we must be very slow to assert its truth. That 
 the capacity is not something superposed upon the sub- 
 stance and capable of being removed from it, cannot be 
 held to be yet demonstrable. 
 
 Kinergety vs. Mass. — The kinergety of substance is 
 clearly a property distinct from all others. Whether it is 
 also a property of matter, or whether it is the consequence 
 of a different property in matter, such for instance as 
 the occupancy of space, need not now be considered (of. 
 Chap. XI. and Part II.). In either case, neither kiner- 
 gety nor any possible cause of it, forms properly any part 
 of the meaning (either denotation or connotation) of the 
 term moss. By almost universal consent, mass denotes 
 quantity of matter, — and nothing further. Por certain good 
 reasons presented in a later chapter, we assume that the 
 quantity of matter — mass — in a body is directly propor- 
 tional to the body's kinergety. But this in no way renders 
 the concept of kinergety a part of the significance of mass. 
 As much should we be justified in assuming that weight 
 and kinergety have anything in common in their meanings 
 because we find them to be proportional to one another in 
 quantity; or that weight and mass are in auy way identical 
 because proportional. This absolute unrelatedness of the 
 concepts expressed by the terms mass and Mnergety, respec- 
 tively, is very generally ignored in the presentation of phys- 
 ics, as is likewise the fact that the proportionality between 
 mass and kinergety (quantity of matter and of capacity for 
 kinetic energy) is an assumption. There is perhaps no 
 point in the elements of physics upon which more con- 
 fusion and haziness of ideas prevails. 
 
 In the following pages it must be remembered wher- 
 ever mass is used, that this is measured only through the 
 assumption of its proportionality to kinergety or to weight. 
 The methods for these measurements will be duly pre- 
 sented. 
 
62 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 Equal Kinergety. — Bodies have equal Mnergety ivhen, with 
 equal velocity, they possess equal quantities of kinetic energy. 
 
 This follows directly from tlie definition of the term. 
 Evidently, then, we may make a second body which shall 
 have a kinergety equal to that of a given constant body 
 which we may call a standard body, by taking from or add- 
 ing to its substance until, by experimental trial with such 
 an apparatus as that already described, it fulfils the require- 
 ment of the foregoing paragraph. 
 
 Graded Set of Kinergeties or Masses. — Having thus con- 
 structed two or more bodies, A, of equal kinergety, we may 
 proceed to make a set of bodies whose kinergeties shall be 
 to one another in any graded series that we may desire. 
 We have, first, to make a single body, B, which when mov- 
 ing with a given velocity, shall be capable of producing the 
 same acceleration upon some constant body as is produced 
 conjointly by two of these bodies A of equal kinergety 
 when moving with the same velocity as B. This new body, 
 B, will have a kinergety twice that of each of the bodies A- 
 Continuing in this way we may produce any multiple kiner- 
 gety, or any desired submultiple. Such a set of bodies of 
 graded kinergeties we have in our ordinary " set of weights " ; 
 although the latter are constructed by a different process 
 which will be described later. Assuming that mass is di- 
 rectly proportional to kinergety, this set of bodies will also 
 constitute a correspondingly graded set of masses and is often 
 so called. 
 
 Measurement of Kinergety. — From the definition of kiner- 
 gety it follows that relative kinergeties are simply propor- 
 tional to the relative amounts of kinetic energy contained in 
 the respective bodies when moving with equal velocities. A 
 primary method^ of measuring relative kinergety of diiferent 
 
 J A primary method is one wliich is based directly on the definition of the 
 quantity to be measured and is in no way dependent on any other mode of 
 measuring the same kind of quantity. 
 
Chap. VI.] KINETIC ENERGY, KINERGETY. 63 
 
 bodies may therefore be laid out along the experimental 
 lines already described. By the spring buffer or other suit- 
 able device, let the body X of which the kinergety is to be 
 ascertained expend all its kinetic energy in accelerating 
 from rest the free motion of a constant body B. Let V de- 
 note the resulting velocity of B. Find by trial what combi- 
 nation of the bodies of the graded set above described will 
 impart to B this same velocity V, the velocity of the com- 
 bination being the same as that of X. The kinergety of the 
 combination is the sum of the kinergeties of its parts, and 
 as these are known, the whole is known, and this is equal 
 to the kinergety of X. 
 
 Although no such method would yield accurate results in 
 practice, because of experimental difficulties, yet no method 
 giving results inconsistent therewith could be accepted. 
 Hence, although " weighing " is the customary practical 
 mode of measuring kinergety (and mass), it is adopted only 
 after rigorous proof that the relative weights of bodies at a 
 given point of the earth's surface are sensibly proportional 
 to their kinergeties, a proof to be presented later. 
 
 A little consideration will show that this primary method 
 for kinergety measurement will also serve for primary 
 measurement of kinetic energy. 
 
 To secure uniformity in measurements of kinergety at 
 different times and places it is obviously necessary to adopt 
 a standard of kinergety. Clearly, the only way to do this is 
 to select some suitable, unchangeable body, and preserve it 
 with the utmost care. Such a standard is afforded by 
 the " International Kilogramme " and by the " Imperial 
 Pound." Quantities of kinergety would be numerically ex- 
 pressed in terms of this standard quantity as a unit,"^ or in 
 terms of some designated multiple or submultiple of it, ac- 
 
 ^ The distinction here indicated between a standard and a unit applies to 
 measurement of quantities of all kinds, and should be carefully observed. 
 Standards of many of the so-called " derived " quantities, such as kinetic 
 
64 FUNDAMENTAL PSTSWAL CONCEPTS. [Part I. 
 
 cording to convenience, e.g. in kilogrammes, grammes, cen- 
 tigrammes, etc., or in tons, pounds, grains, etc. 
 
 Law of Kinetic Energy. — The first proposition under this 
 law is : — 
 
 The kinetic energy of different bodies when moving with 
 equal velocities is proportional directly to their respective kin- 
 ergeties. 
 
 This follows directly from the definition ; since the tin- 
 ergety of a body is defined as being proportional to the 
 quantity of kinetic energy which the body will possess 
 when moving with a specified velocity. The second propo- 
 sition is : — 
 
 The kinetic energy of the same body moving at different 
 times with different velocities is proportional directly to the 
 square of the velocities. 
 
 This proposition requires experimental demonstration. 
 It may be illustrated by the spring-buifer apparatus. Let 
 a given constant body A expend all its kinetic energy in 
 accelerating from rest a second constant free body B. Let 
 this operation be repeated with velocities of A at impact 
 successively in the proportion 1:2:3, etc. Note the corre- 
 sponding velocities of B. Then find by trial with the 
 graded set of bodies above described the kinergeties succes- 
 sively necessary to produce again upon B the various noted 
 accelerations. The velocity of the trial bodies being the 
 same at impact in all trials, the kinergeties requisite will 
 be found to be in the proportion of 1:4:9, etc. But from 
 the first proposition in this section, the kinetic energy of 
 these trial bodies, their velocities having been the same, 
 will have been in the ratio of their kinergeties ; namely, 
 1:4:9, etc. These quantities of energy must have been 
 equal to the respective quantities of energy exerted by A 
 
 energy, heat, electric quantity, magnetic field, etc., are not represented by 
 any material standard, while others are so represented, such as electrical 
 resistance and electromotive force. 
 
CiiAP. VI.] KINMTIC ENERGY, KINEBGETY. 65 
 
 when its velocity was successively in tlie ratio 1:2:3, etc. 
 Hence the kinetic energy of A must have been as 1 : 4 : 9 ; 
 that is, proportional to the square of its velocity. 
 
 Combining these two propositions it follows that the 
 kinetic energy of a body is proportional directly to its kin- 
 ergety and to the square of its velocity. 
 
 The velocity V is the instantaneous velocity along the 
 path of the body, and the corresponding quantity of energy 
 E is the body's kinetic energy relatively to any fixed point 
 on the path, or to any point fixed with reference to the 
 path. The reference point for both path and point must be 
 the same, but may be any chosen point whatever not located 
 upon the body itself. 
 
 If K be used to denote the numerical value of the kiner- 
 gety of a given body (measured as above described), and V 
 to denote its velocity, then by this law the kinetic energy 
 of the body would be proportional to the product KV^. If 
 we were to assume as the unit quantity of energy the quan- 
 tity possessed by a body of unit kinergety when moving 
 with unit velocity, then KV^ would give the numerical 
 value of the kinetic energy of the body in the above case. 
 But for reasons to be stated later, the unit of energy which 
 has been universally adopted in practice has twice this 
 magnitude, so that the factor ^ must be introduced, and the 
 expression becomes ^ KV^ 
 
 Assuming, as we do, that mass and kinergety are directly 
 proportional to one another, then if M denote the mass of 
 the above body, the kinetic energy of the body will also be 
 given numerically by the expression ^ MV"^ 
 
 Primary Measurement of Kinetic Energy. — The method 
 for this purpose would consist in measuring the kinergety 
 of the given body. The body's kinetic energy when mov- 
 ing with any known or measured velocity would then be 
 equal to one-half the product of the kinergety into the 
 square of the velocity. An alternative method would con- 
 
66 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 sist in ascertaining by trial, what combination of the set of 
 bodies of known kinergeties would, when moving with the 
 velocity of the unknown body, have the same amount of 
 kinetic energy. 
 
 The customary scientific unit of kinetic energy is one-half 
 of that possessed by one gramme when moving with a velo- 
 city of one centimeter per second. It is called the erg, but 
 might better be named the kinerg, as suggested in the foot- 
 note on page 60. 
 
 Attention should be here called to the proposition that 
 kinergety is proportional to wei-ghtal, a quantity to be later 
 defined. 
 
CHAPTER VII. 
 ENERGY (Continued). 
 
 Transformation of Energy. — It may be shown experimen- 
 tally by means which are described in text-books of physics, 
 that, — 
 
 Energy may be changed from any one of its forms into any 
 other; that is, may be transformed, either directly or indirectly. 
 
 This is known as the principle of the transformation of 
 energy. Illustrations have been given on pages 25 to 37, 
 and others will be found in the following pages. 
 
 Conservation of Energy. — The foregoing qualitative prin- 
 ciple is supplemented by a quantitative one known as the 
 Conservation of Energy, which may be thus stated : — 
 
 When any quantity of energy of one form disappears, a 
 precisely equal quantity of energy simultaneously appears in 
 some other form or forms. 
 
 These two principles are sometimes united and spoken of 
 as the " Principle of the Transformation and Conservation 
 of Energy." They underlie most of the recent advances in 
 physical science. Their great scope will be obvious when 
 we reflect that every phenomenon is due to a transference 
 or transformation of energy. 
 
 The principle of transformation, as it was historically 
 arrived at, was an inference from experimental evidence. 
 Sundry diverse conditions of bodies were observed under 
 which they had certain powers in common, such as the 
 power to change the state of motion of free bodies, and to 
 move bodies against resistance. These vaguely understood 
 powers were in reality diverse in their nature. Some of 
 them were what we now call forces ; some were what we 
 
 67 
 
68 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 now call energy ; some, even, -were substances. They 
 ultimately came to be designated " The Physical Forces." 
 Each of these "forces" had its own name (p. 53), and 
 at first no common relation was discerned between them. 
 But it was gradually perceived that a relation existed which 
 is now known as transformability. Thus it was discovered 
 that, under certain conditions, motion of bodies could be 
 changed into heat or vice versa; that electrical currents 
 could be made to produce either motion of bodies or heat ; 
 that gravitation could produce and maintain the motion of 
 a body ; and that, if this moving body were kept in motion 
 by gravitation against friction, heat would be developed at 
 the expense of gravitation. As the knowledge of such facts 
 developed, it gave rise to the statement of a principle known 
 as the " Correlation of the Physical Forces." Further study 
 showed that when one of these so-called forces produced 
 another, there was a quantitative as well as a qualitative 
 relation ; so that when a transformation occurred, a definite 
 quantity of one kind disappeared, and a definite quantity 
 of another was simultaneously produced. These quantities 
 were not always the quantities or powers which had been 
 vaguely termed the forces, but were intimate functions of 
 them ; and the quantities which were actually transformed 
 came eventually to be called by the general term energy. 
 
 It is manifest that the establishment of the quantitative 
 law of conservation of energy demands the ability to 
 measure quantities of energy of different forms. If in all 
 its diverse forms, energy were directly recognizable and 
 measurable as a single quantity, the enunciation of the 
 law of conservation would doubtless have been made fully 
 a century earlier. For the general metaphysical idea of 
 the equivalents of cause and effect, and of the impossibility 
 of creating something out of nothing, as well as the denial of 
 the possibility of ^iperpetuum mobile, found acceptance with 
 philosophers at an even more remote period. In fact, how- 
 
Chap. VII.] ENERGY. 69 
 
 ever, not one of the forms of energy is quantitatively meas- 
 urable directly ; each must be obtained as the product of 
 two or more factors of different kinds, which must be 
 independently measured. What the factors were whose 
 products should constitute, for example, gravitation energy, 
 electrical energy, heat, etc. (as we now call them), was far 
 too obscure a matter to be clearly apprehended even by the 
 most penetrating minds, in the state of experimental science 
 preceding the first quarter of this century. Through the 
 progress of experimental science it gradually became mani- 
 fest what the factors were whose product measured that 
 quantity which was constant throughout all its transforma- 
 tions in the various phenomena. 
 
 Thus, it was early recognized that the effectiveness of a 
 moving body, or its power, or what we now call its kinetic 
 energy, was proportional jointly to the mass and the square 
 of the velocity. 
 
 Again, it was found in the case of a body of weight W 
 and mass M, falling freely through a vertical distance H, 
 starting from rest, that its kinetic energy, ^ MV^, was 
 always equal (employing proper units) to the product WH ; 
 V being the velocity at the end of the fall. The quan- 
 tity WH may be called the quantity of gravitation energy 
 exerted, just as ^ MV ^ is the resulting kinetic energy. It 
 was found that when the power of a descending body of 
 weight W was expended against friction, either by stirring 
 water or by rubbing together metal surfaces immersed in 
 water, that the quantity WH was equivalent always to a 
 definite amount of heat, as represented by the product of 
 the weight of the water heated, multiplied into its rise of 
 temperature. 
 
 It was ascertained that in an electrically heated wire 
 whose heat was imparted to a quantity of water, the prod- 
 uct CVT or C^RT was always the equivalent of, or rather 
 was capable of producing, a definite quantity of heat, meas- 
 
70 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 ured as just stated ; where C is the current, V the difference 
 of potential (electrical pressure) between the ends of the 
 wire, E the electrical resistance of the wire, and T is the 
 duration of the current. 
 
 It was discovered that a definite quantity of heat, meas- 
 ured as above indicated, was invariably produced by the 
 combination of definite masses of chemical substances ; for 
 instance, that a gramme of carbon burning in oxygen to 
 form carbon dioxide (carbonic acid) produced a definite 
 number of units of heat. It was found that when a 
 gramme of zinc was completely oxidized, the heat evolved, 
 allowing for incidental operations, was the same whether 
 the action occurred directly or through solution in sulphu- 
 ric acid. But if the latter operation took place in an elec- 
 trical battery, then the quantity of heat generated was less 
 in direct proportion to the amount of energy otherwise ex- 
 pended by the electrical current. 
 
 In these illustrations, gravitation, electricity, heat, and 
 chemical action, would originally have been called " Physi- 
 cal Forces," but according to the modern terminology, gravi- 
 tation is regarded rather as the name of a property exhibited 
 by bodies and due to a form of energy, than as the name 
 of the force corresponding to this property. The latter is 
 ordinarily termed either the force of gravitation (between 
 any two bodies), or gravity (between bodies and the earth). 
 That which is transformed in the above experiment is 
 neither gravity nor force, but the energy of gravitation. 
 Heat is not a force, in the modern view, but a form of en- 
 ergy. Chemical action is now ascribed to a form of energy. 
 Electricity is at present regarded as neither a force nor a 
 form of energy. Whether it shall be regarded as a sub- 
 stance or as a (so-called) fluid, or as identical with the lumi- 
 niferous ether, or whether it should be merely left as one 
 of the things at present wholly unknown, is largely a mat- 
 ter of personal opinion among physicists. The quantity 
 
Chap. VII.] ENERGY. 71 
 
 CVT which is transformed in the above illustration is the 
 energy of the electrical current. 
 
 From this statement it is easy to see how confusion would 
 arise in physical literature as soon as force began to be 
 restricted to its legitimate meaning, but this natural confu- 
 sion was rendered far greater by the failure to perceive, or 
 at least to state distinctly, that force itself is not a kind 
 of energy. Many of the early specific definitions, which 
 moreover are still current, gave it merely the vague charac- 
 ter of " that which " can produce motion. This, unhappily, 
 is a definition not of force hut of energy. 
 
 The principle of the conservation of energy is a generali- 
 zation from experience, and possesses a weight which in- 
 creases as experience continues to confirm the proposition 
 and deductions from it. Our greatest confidence in the 
 principle is established not by the limited number of direct 
 quantitative confirmations, so much as by the concordance 
 between the multifarious applications of the principle and 
 the corresponding observed facts. 
 
 The contradiction of the principle of conservation involves 
 us at once in the assumption that energy can be created out 
 of nothing, since the definition as well as the principle of 
 the conservation of matter (cf. Chap. IX.) shows that 
 energy cannot be a substance, and anything out of which 
 energy can be created would, of course, under the principle 
 of transformation, be classified as energy. But the assump- 
 tion that energy could be produced from nothing would lead 
 to immediate and endless contradiction between facts and 
 the logical deductions from this proposition. It leads di- 
 rectly to the principle of "perpetual motion," which all 
 experimental evidence flatly contradicts. The principle of 
 conservation may then be regarded either as a proposition 
 established by direct experimental evidence, or as one de- 
 duced from the denial of the possibility of perpetual motion 
 — a denial in turn based on experience ; or we may regard 
 
72 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 it as established jointly by both, of these bodies of experi- 
 mental evidence. The first two views have been advocated 
 with undue bitterness in the advancement of claims for per- 
 sonal priority. Laying aside this sort of dispute as in no 
 wise pertaining to science, the third view is doubtless the 
 more rational. 
 
 Criterion for Energy. — From a modern standpoint, the 
 principle of transformation has become largely a matter of 
 definition ; for whatever shape the definition of energy actu- 
 ally takes, if it be a true one it must be consistent with the 
 principle of transformation, and the real criterion to which 
 everything must be submitted before it is allowed to enter 
 the category of energy, is the application of this principle 
 of transformation which may be stated as follows : — 
 
 Aivjtldng is energy which is capable, of being transformed 
 into any liwwn form of energy, or into which any knoicn 
 form of energy can be transformed. 
 
 This criterion can be applied in the following manner, 
 and in fact is always applied when it is desired to test 
 whether or not a quantity or thing is energy. Inspection 
 will at once show that the definition of energy adopted in 
 this volume (p. 20) corresponds to a particular application of 
 this general criterion. 
 
 Starting with one recognized form of energy, as for in- 
 stance kinetic energy, we may proceed to discover a second 
 form by finding something which can be transformed into 
 kinetic energy. We may then discover a third form by 
 finding something which may be transformed into either of 
 the first two, or into which either of them may be trans- 
 formed. Proceeding in this way, the facility of recogni- 
 tion of any new form of energy continuously increases 
 under this criterion. We cannot fail to recognize any form 
 of energy, neither can we include anything which is not 
 energy, for transformability is the one essential test of 
 energy. It may be noted, however, that the principle of 
 
Chap. VII.] ENERGY. 73 
 
 conservatiou does not afflrm that all the energy of any one 
 kind in the universe is convertible into any other form or 
 forms. 
 
 , Historical Sketch. — A brief sketch of the doctrine of 
 energy and of the conservation of energy may serve to in- 
 dicate more forcibly the present dominating position of the 
 latter in physical science. Eor dates and for the names of 
 those physicists who have contributed to this subject — and 
 the list embraces all the leading physicists of this century 
 — reference must be made to such an account as is given in 
 Planck's "Erhaltung der Energie." 
 
 Beginning with the time when it was supposed to be ne- 
 cessary to apply power continuously to free bodies in order 
 to maintain their motion, the progress was slow to the time, 
 two centuries ago, when this was distinctly contradicted and 
 correct views of the law of motion were enunciated by New- 
 ton. But even before that epoch the denial of the possibil- 
 ity of perpetual motion, or more properly of a perpetual 
 motor, had been more or less clearly enunciated; the denial 
 being based partly on a priori arguments, partly upon want 
 of success in experimental trials. Although Newton seems 
 to have had some conception of the idea of conservation in 
 the action of machines, yet the want of anything approach- 
 ing to prominence in his treatment of this subject seems to 
 show that it occupied at least no important position among 
 his views. And he certainly extended it to no part of phys- 
 ics as known in his time, beyond a limited portion of the 
 domain of mechanics. He uses no term equivalent to en- 
 ergy, nor any term, other than a phrase, for mechanical 
 power. Newton's idea of force was very distinctly stated, 
 as shown in part in the definition quoted in Chapter XII., 
 and it corresponded closely with present usage. His meas- 
 ure of force was the time-rate of change of momentum. 
 During the century following Newton's work small prog- 
 ress was made in the development of the concept of energy 
 
74 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 except in the domain of mechanics where the law of conser- 
 vation of vis viva for elastic collision was established, and 
 disbelief in the possibility of the perpetuum mobile was more 
 than once clearly stated. That power was lost (in the sense 
 of annihilated), through friction and other resistances, was 
 still the prevailing belief. But that either power or the 
 motion of matter could be produced out of nothing was for 
 the most part emphatically denied ; still, the state of physi- 
 cal knowledge was such that there was little or no thought 
 of connecting by any common link diverse phenomena such 
 as gravitation, mechanical power, light, heat, electricity, and 
 so on. The end of the eighteenth century, however, saw 
 the substantial beginnings of this broader view, in such far- 
 reaching investigation as that of Rumford and Davy, and 
 the first quarter of the present century witnessed important 
 advances ; not only did the knowledge of physical phenom- 
 ena and laws steadily increase, but the notion of converti- 
 bility, and of the interdependence of phenomena gained 
 strength. Heat, however, was still by many regarded as a 
 form of matter. The nature of light was not understood, 
 and the knowledge of the laws of electricity and magnetism 
 was very incomplete. During the second quarter of the 
 century, the state of physical knowledge was advancing at 
 a rapidly accelerating pace towards the great announcement 
 of the principle of conservation which the middle of the 
 century was to bring forth. The perception of the funda- 
 mental idea of conservation, as well as the first clear state- 
 ment of the principle, was based upon the denial of the 
 possibility of perpetual motion. This denial rested largely 
 upon metaphysical speculation, although supported by an 
 array of experimental evidence confined almost wholly to 
 the domain of mechanics. The extension of the principle 
 of conservation to other portions of physics from such a 
 basis at this epoch, however brilliant as an effort of human 
 reason, and however fully confirmed by subsequent study. 
 
Chap. VII.] ENEBQT. 75 
 
 cannot be considered to have been a rigid physical demon- 
 stration. It was, therefore, not until adequate strictly 
 physical testimony had been afforded through the exhaus- 
 tive experimental work of Joule that the principle of the 
 conservation of energy, although previously formulated by 
 Mayer, could be regarded as a legitimate part of physical 
 science. From the date of the beginning of Joule's work 
 (about 1842), the concept of the nature of energy, and the 
 principle of its transformation and conservation, developed 
 and were confirmed at an ever-increasing rate. Within 
 twenty years, these had progressed to a rank, not second in 
 importance even to the corresponding concept and law of 
 matter. During this short time all laws and phenomena had 
 been subjected to the closest scrutiny as to agreement with 
 the principle of conservation, and immense progress had re- 
 sulted both in the theory of physical science and in the dis- 
 covery of new phenomena. So absolute is the confidence 
 now accorded to the principle, that it is universally adopted, 
 along with the conservation of matter, as the fundamental 
 starting-point in all physical inquiry. 
 
 That the establishment of the principle should have 
 awaited, as it did, a wider knowledge of physical phenomena 
 is easy to see when we consider the complexity involved in 
 the measurement of some forms of energy, and the radical 
 difference in the character of the quantities measured and 
 the phenomena involved. No form of energy is directly 
 measured, as for instance we measure length or weight, but 
 each form is indirectly measured. That is, each form is 
 measured as a product of two or more separately measured 
 factors or quantities. Until these quantities and the means 
 of their measurement were understood, it was impossible to 
 measure the corresponding form of energy. This has 
 already been stated in some detail at page 68. Thus grav- 
 itation energy is measured for objects near the earth's 
 surface by the product of weight into vertical distance 
 
76 FUNDAMENTAL PHYSICAL CONCEPTS. [Fart I. 
 
 moved through. The energy of electrical currents is 
 measured by the product of the current, into the difference 
 of potential between the two points considered, into the 
 time of duration of the current. The relative energy of 
 two charges of electricity is measured by the product of the 
 force into the distance through which the charges are 
 moved towards (or away from) each other, luider that force. 
 Heat is measured by the product of the mass of water 
 heated, into the number of degrees of tempierature through 
 which this rises when the heat is applied to it. Radiant 
 energy is measured by transforming it into heat ; and so on 
 for other forms. The briefest consideration will show that 
 while it might be perceived that the power of a moving 
 body was transformable into heat, or into energy of electric 
 currents, and so on, still any quantitative relation might 
 escape detection for want of knowledge either as to the 
 nature of the quantities to be measured, or as to how they 
 should be measured. And not only was this great obstacle 
 to be overcome, but confusion and delay arose from unfort- 
 unate discussion which involved in a large measure merely 
 questions of terminology. Thus a most bitter controversy 
 was long conducted as to what should be held to be the true 
 measure of force. Descartes and his followers contended 
 that force should be measured by the rate of change of 
 momentum. Leibnitz, strongly supported, held that force 
 should be measured by what we now call the work done, 
 or kinetic energy produced, by its action. The discussion 
 was, however, not one of terminology merely, as might 
 appear from this statement. It involved also the question 
 as to which of these two was the fundamental quantity to 
 be dealt with. In respect to this latter point Leibnitz was, 
 as it now appears, undoubtedly right ; and following him 
 Helmholtz and most of the other prominent physicists deal- 
 ing with this subject up to about 1860 used the term force 
 {Kraft) in a sense entirely equivalent to that in which 
 
Chap. VIL] ENBRGT. 77 
 
 energy is now employed. But ultimately, prolonged usage, 
 supported chiefly by the classic writings of Newton, pre- 
 vailed, and the term force was restricted to its present 
 significance. The term energy, first proposed by Young in 
 1832 as a name for vis viva (MV^), was adopted and is now 
 universally employed to represent what Leibnitz indicated 
 by force (Kraft) as well as the extension of that idea. That 
 this diversity in the significance attached to the word force 
 (Kraft) by most competent authorities should have led to 
 very great confusion in the literature of physics, is most 
 natural. And though there is at present entire unanimity 
 as to the employment of the term, this confusion has not yet 
 disappeared from even recent text-books of physics and still 
 less from colloquial use. 
 
 The commanding position of the doctrine of conservation 
 of energy in physical science can hardly be too strongly 
 insisted upon. It not only furnishes the sure criterion for 
 the discrimination of energy, but it affords the inevitable 
 test to which any proposed operation, or any supposed dis- 
 covery, invention, or law of the physical world must be 
 subjected ; and since all physical phenomena come under 
 its scope, the deduction of these phenomena and their laws 
 from this principle as a starting-point must be possible. 
 Although for nearly forty years the principle has been so 
 firmly established, each year shows almost revolutionary 
 changes in the treatment of the laws of physics, arising 
 from this point of view. Instead of regarding these laws 
 as inductions from observed phenomena, thus passing from 
 the particular to the general, they are more and more 
 treated as deductions from the general principle of the con- 
 servation of energy. And fruitful as the principle at first 
 was in correlating phenomena and promoting discovery, it 
 has now proved hardly less effectual in unifying and 
 generalizing the presentation of these laws. 
 
 Other Statements of the Principle. — The principle of con- 
 
78 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 servation may be also stated in two other forms as follows. 
 First, in any system of bodies entirely enclosed by a boun- 
 dary, real or imaginary, through which no energy passes 
 during the time under consideration, the total quantity of 
 energy remains constant whatever phenomena take place 
 within the system. Second, the total quantity of energy in 
 the universe is a constant. This second statement is to be 
 regarded as merely a special case of the first, since the uni- 
 verse, whether finite or not, comprises everything, and can 
 therefore neither be receiving nor giving out energy. Both 
 of these forms are open to criticism in ways from which the 
 statement at page 67 is free. 
 
 Examples of the Teansfoemation of Exeegt. 
 
 Let us now take a few specific instances of transformation 
 and conservation of energy. 
 
 Gravitation and Kinetic Energy. — Kinetic energy can be 
 transformed into gravitation energy by directing the motion 
 of a body upwards from the earth's surface. Thus if a body 
 be thrown upwards from the ground, it will continue to rise 
 with retarded motion until its velocity becomes zero, that is, 
 until its total kinetic energy has disappeared. If this is 
 done under such conditions that no other changes take place 
 except such minor disturbances or losses as may be allowed 
 for, what becomes of the kinetic energy ? If the body be 
 now allowed to fall freely under its own weight, it will have 
 acquired, when it has descended through exactly the same 
 distance S through which it ascended, a velocity V, identi- 
 cal with that with which it started upwards. It will then 
 have received exactly the amount of kinetic energy with 
 which it started upwards. Whence has this energy come ? 
 
 This experience is corresj)Ondingly true of the gravitation 
 between any two portions of substance ; the body and the 
 earth are merely convenient examples, the more so because 
 
Chap. VII.] ENERGY. 79 
 
 of the great mass of the earth which renders its accelera- 
 tion in such cases negligible. What inference are we to 
 draw ? In falling, the object acquires energy of a definite 
 amount and at the same space-rate. In ascending, the ob- 
 ject gives up energy equal in amount and (as may be shown 
 from its uniformly retarded motion) at an exactly equal 
 space-rate. We know also (p. 44) that during the ascent, 
 as well as during the fall, and at all other times, there is a 
 force exerted downwards upon the body ; namely its weight. 
 This continual and uniform action of weight is indicative 
 of a store of energy always acting upon the body to produce 
 this weight. The exact and invariable relation between the 
 amount and rate of energy in ascending and descending, are 
 inconsistent with any hypothesis which does not assume 
 that the energy given out during the ascent is stored up 
 and an equivalent imparted during the descent. It is natu- 
 ral to assume that the energy given up in ascent is merely 
 imparted to this store, and that in descent an equivalent 
 amount is returned from that store ; for the body would 
 give up no energy in rising if not acted upon by the energy 
 which causes its weight, that is, if it were not moving in oppo- 
 sition to a force ; since it is not accelerating any knovm por- 
 tion of matter. 
 
 It is important to notice the evidence for this store or 
 stock of energy. The weight of the body is precisely the 
 same in its lowest as in its highest position, that is, before 
 the kinetic energy has been converted into energy of gravi- 
 tation. It is, therefore, not this newly formed gravitation 
 energy which is producing the weight ; and further since 
 this added gravitation energy fails to give the slightest evi- 
 dence of its existence by increase of the weight of the body, 
 it must be an insignificant amount as compared to that store 
 of gravitation energy which is producing weight. Again, 
 a body at a point above the earth's surface will acquire 
 energy in descending whether it has previously been in a 
 
80 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 position nearer to the surface or not. To take an extreme 
 illustration, a meteorite undoubtedly follows the same law 
 of approach to the earth as any equivalent portion of terres- 
 trial matter would if raised from the surface to the position 
 of the meteorite. That is, the preliminary transformation 
 of other energy into gravitation energy by raising a body 
 from the earth's surface to a more elevated position is not, 
 so far as we know, a preliminary requisite to the capacity 
 of that body to acquire energy in falling. To put this in a 
 more general way, we may say that so far as we have any 
 knowledge of the state of the universe anterior to the pres- 
 ent, this knowledge indicates that the average distance 
 between tlie particles of matter forming the earth has been 
 progressively lessening. This, apart from pointing to the 
 long existence of the phenomenon of gravitation, shows that 
 the energy which gives to bodies a tendency to approach 
 the earth's surface preceded the present state of aggrega- 
 tion of the materials of the earth. Therefore it is not 
 necessary to assume that in order for bodies at a distance 
 from one another to have a tendency to approach, they 
 should previously have been situated in a, position of closer 
 proximity. 
 
 The conception of the existence of a store of gravitation 
 energy beyond the insignificant amounts arising from change 
 of position of bodies is thus a reasonable inference. We 
 may naturally inquire where this store of energy resides, and 
 might seek it first in the body itself. But, we have not the 
 least indication, through any change whatever, that the body 
 is in any way modified by its removal upwards from the 
 earth's surface. We should most naturally look for evi- 
 dence of the existence of the stored-up energy, as the body 
 rises, in increased motion of some kind in its particles ; but 
 motion of the particles of bodies produces several well-known 
 phenomena, temperature, radiation, etc., and none of these 
 are in the least modified. However closely we scrutinize 
 
Chap. VII.] ENERGY. 81 
 
 the body, there is no indication to be found either of the 
 presence of the energy producing weight, or of the added 
 energy which has been stored up through the increased ele- 
 vation of the object. 
 
 Does the store of gravitation energy then exist outside of 
 bodies, and is it perhaps distributed throughout space ? 
 This seems the only natural alternative hypothesis and is 
 one which the mind very readily accepts. Still, beyond the 
 facts of gravitation, there is not the least indication of the 
 presence of such a universal store. 
 
 For a complete understanding of the treatment of the 
 doctrine of energy, it is not necessary to choose between 
 these hypotheses or to adopt any view whatever. The first 
 of the hypotheses involves us in a further complication of 
 accounting for the action of the energy resident on one body 
 upon the energy resident on the other body at a distance 
 from the first ; since " action at a distance," without the 
 intervention of some intermediate material agency, so vigor- 
 ously disclaimed by Newton and by Faraday, continues to be 
 regarded as an vmtenable supposition. We are obliged to 
 infer some medium for the transmission of gravitation force. 
 Under the hypothesis that the store of gravitation energy is 
 uniformly distributed throughout space, no such supposition 
 is necessary, so that the second hypothesis has perhaps the 
 merit of greater simplicity. It is undoubtedly the one 
 which more nearly represents the general trend of scientific 
 speculation (cf. Part II.). It is, however, of the utmost 
 importance to recognize the possibility of more than one 
 hypothesis as to the locality of this energy. We must give 
 ourselves the broadest latitude possible, and the greatest 
 freedom from the bias which would result either from 
 adopting an insufficient hypothesis, or from failing to per- 
 ceive the possibility of an alternative. 
 
 If we bear in mind that this relation holds between any 
 two bodies, at least within the range of our solar system, we 
 
82 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 must infer that this store of energy, if such there be, must 
 extend at least to tlie boundaries of this system. The ex- 
 istence of companion stars, apparently in orbital motion 
 about each other, indicates the probability that gravitation 
 also attends some other portions, at least, of the universe. 
 We have, however, no direct evidence that gravitation ex- 
 tends throughout all portions of space, though it is perhaps 
 difficult to imagine that its extent is limited. 
 
 To summarize briefly then we may say that we may infer 
 the existence of a store of what we choose to call gravitation 
 energy, perhaps located upon bodies, perhaps distributed 
 throughout space, whose extent is at least as great as that 
 of the solar system, and which acts continuously and uni- 
 formly on all portions of substance. This action produces 
 between any two bodies a tendency to acceleration towards 
 one another, such that if the portions are free and start from 
 rest, they will be made to approach each other with acceler- 
 ated motion, in a straight line joining their centers, and each 
 will acquire energy relatively to its former condition of rest 
 at a definite space-rate, which will be the same for both. 
 That is, the energy exerts upon each of the two bodies an 
 equal force, tending to make them approach each other 
 along a straight line. This force is what we call the force of 
 gravitation ; or when the two bodies are respectively an ob- 
 ject and the earth, we call the force, the weight of the object.' 
 This force is constant for any two given bodies, at any 
 given distance from each other, at any part of space (at least 
 within the solar system) ; but depends upon the bodies and 
 their distance apart, according to Newton's well-known law 
 
 1 More strictly, the action of gravitation between an object and tlie 
 earth is called the force of gravity, or merely the gravity of the object. 
 Weight is the force of gravity, diminished by the effect of the earth's rota- 
 tion, the object being supposed to rotate with the earth. In other words, 
 weight is the force of gravity minus tlie so-called centrifugal force arising 
 from the body's rotation with the earth. 
 
Chap. VII.] ENERGY. 83 
 
 that the force is proportional to the product of the masses, 
 and inversely as the square of the distance between the 
 centers of the bodies. 
 
 Gravitation and Heat. — Suppose some heavy " weights " to 
 be suspended by a cord passing over pulleys and wound 
 around a vertical drum as in Joule's experiment (see any 
 text-book on physics). Suppose this drum to turn on a ver- 
 tical shaft which extends downward into a closed cylindrical 
 box containing water, and to have paddles projecting later- 
 ally from it inside the box. Suppose the box to be securely 
 held to prevent turning and the weights to be allowed to 
 descend; the water will be violently stirred; gravitation 
 energy is imparted to the weights ; they descend and through 
 them the gravitation energy is transformed into energy of 
 elasticity in the cord and as such transferred through the 
 cord, the shaft, and the paddles, to the water. It is there 
 transformed into kinetic energy of eddies in the water, 
 which is in turn transformed into heat through the friction 
 of the water upon itself and upon the paddles and the walls 
 of the cylinder. All losses being allowed for, the gravitation 
 energy expended (weight x distance of descent) is the equiv- 
 alent of the heat produced (mass of water x number of de- 
 grees of the rise of temperature, with allowance for heat 
 capacity of the box). 
 
 Kinetic Energy into Heat. — The total gravitation energy 
 imparted to the "weights" in the foregoing experiment, 
 during their steady slow descent, is equal to the kinetic 
 energy which they would have acquired in a free fall 
 through the same distance. By a suitable device, the 
 weights might have been allowed to fall freely through the 
 same height as before, and the kinetic energy thus ac- 
 quired might have been made to drive the paddles in the 
 machine of the foregoing experiment. The heat generated 
 by the total expenditure of this energy would be found to 
 be precisely the same as before. 
 
84 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 In a collision between a bullet and a target (taken as the 
 type of the collision of an inelastic substance like lead, 
 with a resisting body), the kinetic energy of the bullet is 
 largely transformed into heat. This arises through the 
 friction of the parts of the bullet upon each other as it 
 becomes distorted, through the friction of the bullet upon 
 the target in penetrating it, and through friction of parts 
 of the material of the target upon each other. In such a 
 case, however, a considerable portion of the energy is 
 changed into other forms than heat. This is also true in 
 most unconstrained transformations. 
 
 The transformation of kinetic energy into heat is going 
 on at almost every point in the universe where motion 
 exists ; since it takes place wherever motion is in progress 
 against f rictional resistance, and friction is almost universal 
 in its occurrence. 
 
 In a descending clock weight, gravitation energy is 
 transformed into heat through the friction of the moving 
 parts of the clockwork and by the air currents set up by 
 the pendulum and the moving wheels. Some of it is trans- 
 formed into energy of sound vibration which, in turn, is 
 broken down into heat energy through the impact of the 
 vibrations upon inelastic substances or from transmission 
 through them. 
 
 Electrical Energy into Kinetic Energy. — Two pith balls 
 electrified with positive charges tend to recede from one 
 another ; in other words, if free to move they will be ac- 
 celerated away from each other. Some form of energy, 
 which we call electrical energy, has been brought into 
 action by this process of electrification, and is exerting a 
 force upon each of the balls, tending to separate them. 
 "When they are free to move, this electrical energy must, 
 under the principle of conservation, be transformed into 
 the kinetic energy resulting. The charge of electricity on 
 either body does not diminish however far apart the two 
 
Chap. VII.] ENERGY. 85 
 
 may be separated, as can be shown by the ordinary tests 
 for charge ; but the power of the charges to produce kinetic 
 energy diminishes with the distance between the balls. 
 The electrical energy can be experimentally shown to 
 diminish in the proportion as kinetic energy is produced. 
 
 If the pith balls are not free to move, but are opposed by 
 frictional or other resistance so that they can move with but 
 a limited velocity, the electrical energy is transformed, not 
 into kinetic energy, but into such other forms as heat if the 
 resistance is due to friction, or gravitation if the resistance 
 is due to weight, and so on. 
 
 Electrical Energy into Heat. — When a current of electric- 
 ity is flowing through a conductor, it produces heat in that 
 conductor at a time-rate CV, where C is the current flowing 
 in the conductor and V is the difference of the electrical 
 pressure at the ends of the conductor (difference of poten- 
 tial) causing this current. Heat is generated in proportion 
 to this product CV. Hence under the principle of conser- 
 vation, this product is the measure of the electrical energy 
 expended. In just what this energy consists we do not 
 know ; but of its existence besides the production of heat, 
 we have evidence in the production of kinetic energy, and 
 in the maintenance of motion against resistance, as will be 
 shown in a later paragraph. 
 
 Chemical Energy into Heat. — If a piece of carbon be 
 brought into contact with oxygen at a sufliciently high 
 temperature, the familiar phenomenon of combustion oc- 
 curs, that is, the carbon combines with the oxygen produ- 
 cing carbon dioxide, and the operation is attended by the 
 evolution of heat. Similarly zinc in combining with oxy- 
 gen evolves heat ; and in general the direct chemical com- 
 bination of two or more elements likewise produces heat. 
 Since heat is produced, that is, a form of energy is made 
 to appear, some energy must have disappeared, and this is 
 the energy which gave to these substances their tendency to 
 
86 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 enter into chemical combination. As has been stated, this 
 may conveniently be called chemical energy. 
 
 This form of energy is even more obscure as to its nature 
 than gravitation energy, and in a certain sense is more com- 
 plicated i n character ; for it apparently exercises a discrim- 
 ination between portions of matter, which is not dependent 
 merely upon their masses but upon the kinds of substance of 
 which they consist. Thus when carbon and oxygen unite, 
 the ratio between the mass of carbon and the mass of oxy- 
 gen in the combination is always a definite one. Also the 
 quantity of heat generated per unit mass of carbon entering 
 into combination, or per unit mass of carbon dioxide result- 
 ing from the combination, is a definite amount always the 
 same under the same external conditions. Similarly, hydro- 
 gen combines with oxygen in definite proportions by mass ; 
 hydrogen with chlorine ; and so on throughout a long range 
 of direct combinations of elements. In each case a definite 
 quantity of heat is produced in the combination per unit 
 mass, reckoned in terms of either constituent or of the re- 
 sulting compound. The total quantity of heat is in direct 
 proportion to the mass of the compound. 
 
 This heat of combination aj)pears to be the equivalent of 
 all or nearly all of the chemical energy which caiises the 
 union, unless some change such as increase or diminution of 
 volume or the produ.ction of electrical current, attends the 
 direct combination. In such instances the sum of all the 
 sundry quantities of energy produced is equal to the heat 
 which would be produced by the same combination if heat 
 alone resulted. The heat produced, with due allowance for 
 such external actions as attend expansion, etc., is taken 
 therefore as a measure of the chemical energy which is 
 brought into play in the combination. 
 
 Speculations as to the nature of chemical energy and the 
 selective properties of the elements have been but scant- 
 ily successful with the remarkable exception of Professor 
 
Chap. VII.] ENERGY. 87 
 
 J. J. Thomson's development of the Vortex-atom Theory 
 (Part II., Chap. IV.). Some further, though crude, sugges- 
 tions are briefly given in Chapter V. of Part II. 
 
 The above instances are taken as merely the simplest ones 
 typical of this conversion of energy. The reactions between 
 chemical compounds involve, in addition to simple transfor- 
 mation of chemical energy into heat, other transformations 
 more or less complex, which render them less well suited 
 for elementary illustration than the direct combination of 
 elements. 
 
 Chemical Energy into Electrical Energy. — - The energy 
 which in the foregoing section was shown to be trans- 
 formable into heat, is under suitable conditions convertible 
 directly into electrical energy. Perhaps the most familiar 
 illustration is that of the simple zinc-copper-sulphuric acid 
 cell. This consists of a jar containing dilute solution of 
 sulphuric acid, into which, and not in contact with each 
 other, dip a piece of zinc and a piece of copper. If these 
 two metals be connected by wire outside of the jar, a cur- 
 rent of electricity will flow in the wire and through the 
 cell, and will produce heat (although at a very slight rate 
 since the amount of energy expended in the wire is small). 
 The chemical action going on in the cell may be regarded 
 as merely the oxidation of zinc, although the operation is 
 attended with the evolution of hydrogen gas together with 
 the production and solution of zinc sulphate. If the total 
 electrical energy be measured in the usual way by measur- 
 ing the total current and electrical pressure in the circuit, 
 this will be found an exact equivalent for the heat which 
 would be produced by the direct oxidation of zinc, with due 
 allowance for the minor transformations attending the solu- 
 tion of the zinc sulphate, the evolution of the hydrogen, and 
 the reduction of volume of the oxygen when it combines 
 with zinc in the direct oxidation. Thus, by means of suita- 
 ble processes, chemical energy may be transformed directly 
 
88 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 into electrical energy; and this in turn may be changed 
 directly into heat, or by a motor into mechanical energy or 
 other forms. 
 
 Magnetic Energy into Kinetic Energy. — This conversion 
 is familiarly illustrated by the motion of a free piece of 
 iron towards a magnet. Moving under the influence of the 
 magnetic energy of the magnet, the iron acquires increasing 
 amount of kinetic energy. This action of the magnet is so 
 closely related to the action which accompanies a solenoid 
 carrying a current of electricity, that an intimate relation 
 between electrical and magnetic energy is at once suggested, 
 and this has led to an electrical theory of magnetism which 
 regards the two energies as fundamentally the same, mag- 
 netic energy being merely a manifestation of electrical 
 energy. The object here is merely to point out its 
 existence. 
 
 Heat into Mechanical Energy. — We are familiar with the 
 series of conversions of energy which take place in the 
 ordinary production of mechanical energy from coal through 
 the steam engine. They may be enumerated as follows : — ■ 
 transformation of chemical energy of coal and oxygen into 
 heat energy, when the coal is burned under the boiler ; the 
 transformation of some of this heat into other forms of 
 energy in producing steam from water, as well as the stor- 
 ing up of heat energy as such in the steam ; of this total 
 energy of the steam, a part is converted into mechanical 
 energy as the steam is expanded behind the piston of the 
 steam engine, driving the piston forward against the resist- 
 ance of the machinery. In this process, during the expan- 
 sion of the steam, some heat is transformed into mechanical 
 energy. 
 
 A similar illustration consists in enclosing a quantity of 
 compressed gas under a piston loaded with a suitable weight, 
 and allowing the gas by its expansive pressure to force this 
 piston backwards against the resistance afforded by this 
 
Chap. VII.] ENERGY. 89 
 
 weight, thus producing gravitation energy. This will be 
 done at the expense of the heat energy contained in the 
 gas, as will be indicated by a lowering of its temperature. 
 
 Neither of these illustrations constitutes anything like a 
 demonstration that heat can be converted into other forms 
 of energy. Such a demonstration of a perfectly rigid char- 
 acter can be given by use of the Carnot cycle and imagi- 
 nary heat engine. Demonstration may be found in text- 
 books of physics, and is clearly expressed in Tait's " Heat " 
 and Maxwell's "Theory of Heat." 
 
 Electrical Energy into Mechanical Energy. — If a steady 
 current of electricity be sent through an electric motor, and 
 the losses in the motor duly allowed for, it will be found 
 that the motor is capable of producing kinetic energy of an 
 amount equivalent to the product CVT, where C is the cur- 
 rent (supposed to be constant), V the difference of electrical 
 pressure at the terminals of the motors, and T the duration 
 of the experiment. If the motor is not made to produce 
 kinetic energy directly, it can be made, as it usually is, to 
 maintain motion of machinery against resistance such as 
 friction, weight, etc., thus producing energy of other recog- 
 nized forms. The quantity of energy thus furnished will 
 also be found to be in direct proportion to CVT. Thus in 
 either case we have the product CVT producing an equivar 
 lent of some form of energy, so that it must correspond 
 under the law of conservation to the quantity of electrical 
 energy expended. 
 
 We also find that, when the energy produced by the cur- 
 rent corresponds in quantity to the heat produced in the 
 wire, the current is no longer capable of producing other 
 energy ; or if other energy is being produced simultaneously 
 in proportion to CVT, it is no longer capable of producing 
 this amount of heat. If both heat and mechanical energy 
 are being produced by the current, then the quantity CVT 
 must be proportional to the sum of the energies of the two 
 
90 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 kinds produced. These various facts not only confirm the 
 inference that a current of electricity possesses electrical 
 energy, or flows in virtue of electrical energy, but also the 
 inference that the product CVT measures the energy of a 
 steady current. 
 
 Remarks on the Transformation of Energy. — The various 
 statements made in this section are necessarily based on 
 hypotheses of one or another kind, and are therefore to be 
 distinctly regarded as suggestive rather than descriptive. 
 
 "When a free body is rooved by gravitation energy, the 
 transformation of that energy into the kinetic energy of the 
 body appears to be a direct one. Gravitation seems to act 
 directly upon the substance of the body without the inter- 
 vention of any other form of energy ; but this appearance 
 may be illusory. Similarly when the kinetic energy of a 
 body is transformed into energy of gravitation, the pro- 
 cess is also direct. But whether gravitation acts directly 
 upon the matter of which the body is composed, or whether 
 it acts only upon the aggregates of matter which we call 
 atoms, or upon the more indefinite aggregates which we call 
 particles, is not known. We know gravitation only through 
 its association with bodies. That is, we know it as a prop- 
 erty of bodies, and the only hypothesis of any weight re- 
 specting its nature locates it as acting upon the atoms from 
 without. Neither of the existing hypotheses (Le Sage ; 
 Vortex-atom) indicates that it acts upon the matter of 
 which the atoms are composed, and by. the vortex-atom 
 hypothesis the differentiation of matter into atoms is a 
 necessary antecedent of gravitation. In these hypotheses 
 the action of gravitation energy is, however, not direct 
 but through intermediate conversion into energy of 
 elasticity. 
 
 The transference of kinetic energy from one elastic body 
 to another in collision, although at first sight seemingly a 
 direct process, really involves a double transformation. All 
 
Chap. VII.] ENERGY. 91 
 
 the energy transferred is transformed, in the process, into 
 energy of elasticity ; transferred as energy of elasticity, 
 from particle to particle, and from one body to the other, 
 and is retransformed into kinetic energy in the second body. 
 This may be most simply traced out in imagination in a 
 collision of two perfectly elastic spherical balls of equal 
 mass and of homogeneous structure. 
 
 The transfer of heat energy from one body to another, or 
 from particle to particle of the same body, is supposed to 
 involve the same sort of transformation into elastic energy 
 and back again into kinetic, as occurs in the above instance 
 of collision of elastic bodies. This of course is based on 
 the supposition that heat is the kinetic energy of random 
 vibration of the molecules or particles of bodies. 
 
 Transformation of heat energy into other forms, through 
 the motion imparted to a body (e.g. a piston), by an expand- 
 ing gas, is likewise double. It involves the transference of 
 the elastic energy of the gas into elastic energy in the body 
 against which it presses, and the transference of this elastic 
 energy from particle to particle of the body and its final 
 transformation into energy of motion of that body, or into 
 such other form as may be the result of the process. The 
 energy of elasticity of the gas is ordinarily ascribed to its 
 heat energy, so that if we go back a step further in the 
 process we have, — first, the transformation of the heat 
 energy into elastic energy, that is, the transformation of 
 kinetic energy of the molecule into elastic energy as the 
 molecule strikes the side of the body acted upon ; second, 
 the transference of that elastic energy to the particles of 
 that body, and so on. 
 
 This intervention of elasticity, by which other energy is 
 transformed into elastic energy, and thence into some other 
 form, seems to be universally present as an intermediate 
 step in all energy transformations. But this fact is so 
 commonly overlooked as to sometimes render it a surprise 
 
92 FUNDAMENTAL PHYfilCAL CONCEPTS. [Part I. 
 
 ■when stated. Indeed, few text-books even name elasticity 
 among the forms of energy. The oversight may be ac- 
 counted for largely upon the ground that the double trans- 
 formation usually involves little loss or dispersion of energy, 
 so that the omission introduces no conspicuous error. 
 
 The presence of elastic energy and the transformations 
 of it in structures under strain, determine the forces in the 
 members of such structures ; and much aid might be ren- 
 dered to students in the study of statics of structures by 
 the consideration of this energy and its transformations. 
 
 Further Remarks on the Definition of Energy. — This ability 
 of energy to transform itself seems at first sight beyond our 
 original definition, but it may not prove to be so. For, if 
 all energy should prove to be in reality kinetic (cf. Part II.), 
 every transformation is but a redistribution of motion 
 among bodies or parts of a body. Thus a transformation 
 of molar kinetic energy into heat may be only a change from 
 molar to molecular kinetic energy. If gravitation energy 
 is due to some mode of motion of special particles of sub- 
 stance as in the Le Sage and Yortex-atom hypotheses, the 
 transformation of this into kinetic energy is again merely 
 a redistribution of kinetic energy. Like remarks would 
 apply to the other forms. 
 
 Exertion of Force by Energy. — It may not be superfluous 
 to direct attention again to the fact that the definition of 
 force does not imply that energy is at all times exerting 
 force, or that all forms of energy are capable of exerting 
 force. Whether or not the latter is true, is impossible to 
 determine in the present state of knowledge. Inspecting 
 all possible transformations of energy one after another, it 
 appears upon the surface that all forms of energy do exert 
 force. Looking more closely, however, we find that, in 
 cases where the nature of the action which we call force is 
 best known, the force is exerted only through the interven- 
 tion of elastic energy. And, in cases where we are obliged 
 
Chap. VII.] ENERGY. 93 
 
 to resort to hypotheses to explain the action, the only ac- 
 ceptable hypotheses assume the existence of elastic energj"-, 
 which is so called into play as to exert the force. For ex- 
 ample, in the transference of kinetic energy through colli- 
 sion of bodies, we might at first sight say that the kinetic 
 energy exerted force, but upon examination, we find that 
 the kinetic energy of the acting body is certainly trans- 
 formed into elastic energy, which exerts force between the 
 parts of this body, and between this body and the one acted 
 upon. This is the case of transference of energy which we 
 best understand. In this the only force involved, so far 
 as we know, is that exerted by elastic energy. The process 
 is not a direct transference of kinetic energy, but involves 
 an intermediate transformation into and out of energy of 
 elasticity. Again, we say that gravitation energy exerts a 
 force ; but, when we attempt to explain how it exerts force, 
 we are obliged to resort to hypothesis, and every explana- 
 tion of this kind yet offered, involves the action of elasticity 
 and ascribes the exertion of force to this elastic energy. 
 
 Accustomed as most students are to regard heat energy 
 as kinetic, heat is perhaps usually thought of as not produ- 
 cing force. But if we analyze this view, we shall see that it 
 results from hypothesis and is not a direct statement of the 
 facts. Suppose any substance whatever, for instance a solid 
 body, to be enclosed by an unyielding boundary. Let heat 
 be imparted. The temperature will in general rise, by which 
 we recognize that a portion at least of the imparted heat 
 remains as heat in the substance. At the higher tempera- 
 ture, the substance will exert a pressure, — a force, — and 
 usually a very great one against the retaining boundary. 
 This force so far as we know is due to (is the action of) the 
 heat energy. It is due to that portion of the heat which 
 remains as heat, and not to any portion which has been 
 transformed. If, therefore, we lay aside all hypothesis we 
 are compelled by the facts to say that, in this example at 
 
94 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 least, heat exerts force. If, on the other hand, we choose to 
 adopt the hypothesis that heat is molecular kinetic energy 
 and that this pressure arises from the bombardment of the 
 boundary by the elastic particles of the substance in their 
 heat vibration, then we transfer the exertion of force from 
 heat energy to elastic energy. Thus, luitJiout hypothesis, we 
 must assume that heat exerts force ; ivitJi hypothesis we again 
 revert to elastic energy as the ultimate source of force. 
 
 While we must not fail to discriminate sharply between 
 propositions deduced directly from observation and those 
 based upon hypothesis ; still in view of the apparently uni- 
 versal production of the elastic energy as an intermediate 
 step in all energy transformations, and in the exertion of 
 force, we can hardly avoid surprise at the scant recognition 
 accorded to this form of energy. 
 
 Dissipation of Energy. — In stating the principle of con- 
 servation, the reservation was made that the principle did 
 not declare that all the energy of one kind could be trans- 
 formed into energy of other kinds. Of the line of facts to 
 which this relates only the briefest mention need here be 
 made. 
 
 In almost all transformations, whether natural or artifi- 
 cial, some portion of the original energy is changed into heat 
 through the overcoming of frictional and other resistance. 
 There is thus a continuous tendency to increase the amount 
 of heat energy of the universe at the expense of other forms, 
 and this tendency is not offset, so far as known, by any 
 reverse process. Moreover in the transformation of heat 
 energy into other forms, there is a continuous loss of avail- 
 ability of heat energy. Thus it is shown to be a natural 
 law that even the most efficient heat engines, taking heat 
 from a source at a given temperature and working between 
 that temperature and a lower one, — as all heat engines must 
 work, — can convert into other forms of energy only a defi- 
 nite and limited fraction of the heat taken up ; the remain- 
 
Chap. VIL] ENERGT. 95 
 
 der of the heat energy beiug given out in the form of heat 
 at the lower temperature. Also it is impossible to derive 
 any power from heat by cooling a body below the coolest of 
 surrounding objects. 
 
 Again in every heat transformation there is a greater or 
 less accidental escape of heat ; further there is a continual 
 flow of heat by conduction, radiation, and convection, from 
 sources at a higher temperature to points at a lower tem- 
 perature. 
 
 Thus it is shown that there is first, a continuous trans- 
 formation of other forms into heat ; and second, a continu- 
 ous lessening of the availability of heat by the approach of 
 all bodies to a common level of temperature. Should such 
 a condition ultimately be reached, there is no recognized 
 way by which this mass of unavailable heat could be retrans- 
 formed into the other forms of energy. Thus it appears 
 that the universe maybe approaching an ultimate condition 
 very different from the present in respect to the diffusion 
 of energy and the consequent condition of matter. 
 
 This progressive change in energy is called the dissipa- 
 tion or degradation of energy. That there may be in nature 
 some compensating process, either continuous or saltatory, 
 is not denied, and speculations upon this topic are possessed 
 of great interest. 
 
CHAPTER VIII. 
 
 FORCE MEASUREMENT. 
 
 Quantity of Force. — The amount of a force may be prac- 
 tically measured in various ways, but amount or quantity of 
 force must be first defined in a mathematical ^yay consistent 
 with the definition of force itself. Since force is "that 
 action of energy by which it produces tendency to change 
 in state of motion of bodies," since this change involves the 
 acceleration of the body if free and therefore the imparting 
 of energy to it as referred to its former state of motion, 
 force is naturally measured as the rate at which energy is 
 thus imparted. Now it is a matter of observation that 
 when a body moves under this action of energy, it receives 
 energy at a definite space-rate at every point of its path, 
 independently of the epoch or the velocity at which it 
 traverses that path, except in special cases. That is, if we 
 fix our attention upon any given point of the path, then the 
 amount of energy which the body will there receive in 
 traversing any specified distance along the path will always 
 be the same whether the motion occurs at one time or 
 another, and whether the velocity of the body at the time 
 is great or small. 
 
 If the action be uniform, the force is called constant, and 
 the space-rate of transfer of energy to the body will be the 
 same at all points of the path. Let E represent the total 
 energy imparted to the body in traversing a distance S 
 along the path, then the space-rate will be E/S, and the 
 force will have the constant value 
 
 F = E/S. 
 
Chap. VIII.] FOHCE MEASUREMENT. 97 
 
 In such a case, to measure F we might measure E and S. 
 Their quotient would give the value of F. If the action is 
 not uniform, that is, if it is not the same at all points of the 
 path ; and if as before, E represents the total quantity of 
 energy received by the body in traversing the total distance 
 S; then E/S will represent the average space-rate, and F 
 will represent the average value of the force. The force 
 in this case is said to be a varying force, — varying with re- 
 spect to the value of the force along the path and not to 
 the time of its action. In the case of a varying force, if e 
 represents the quantity of energy received by the body in 
 traversing a very small length s of the path, then the aver- 
 age value of the force over this small element s will be 
 /= e/s. As s is taken successively smaller and smaller, 
 the limit toward which the ratio e/s approaches is more 
 and more nearly the actual value of the force /at the point 
 of the path where s is taken ; and this limiting value (de- 
 noted in the calculus by de/ds) is the actual value of / at 
 that point. 
 
 A primary method for the measurement of force may, 
 therefore, consist in measuring the space-rate of transfer- 
 ence of energy when this force is exerted. 
 
 When a body moves freely under a force, it moves with 
 accelerated motion. If the force be constant, the motion is 
 uniformly accelerated. It is consistent with the definition 
 of force to measure it in terms of this acceleration and of 
 the kinergety (or mass) of the body accelerated. Force 
 may be thus measured by the product MA, where M is the 
 kinergety or mass of the body, and A is the rate of accelera- 
 tion at any given instant, or point of the path. This method 
 of measurement will be discussed in a later paragraph. It 
 is in some respects less satisfactory, and is less simple than 
 the measurement as a space-rate of energy. The latter 
 method connects force directly with energy, and brings into 
 account the fact that this action of energy is connected 
 
98 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 directly witli space and not with time. It has the special 
 advantage of applying to all motion whatever, whether free 
 or constrained ; whereas the time-rate measurement applies 
 only to freely moving bodies. 
 
 To ascertain the average value of a force F, we have then 
 to measure the energy E transferred to the body while mov- 
 ing over the distance S. If the energy transferred appears 
 wholly in the kinetic form, as it does when the body has 
 entire freedom of motion, then E = |- MV^ ; so that E be- 
 comes known, if we know the mass M of the body and its 
 velocity V acquired while moving freely under the force 
 through the space S, starting from rest. Thus using the 
 centimeter, gramme, and second, as units of length, mass, 
 and time, respectively (C.G.S. system), suppose a body hav- 
 ing a mass of 5 grammes (= ]M) to start from rest and to 
 move freely through a distance of 200 cm. (= S) under the 
 influence of a force F whose average amount it is desired to 
 measure. Let its velocity at the end of this space be 40 cm. 
 per second (= V). Then E = i MV = ^ x 6 x 40^ = 4000 
 
 ergs. The average force therefore is F = — = = 20 
 
 dynes. ^ 200 
 
 If the motion is not free, then some of the imparted energy 
 does not appear in the kinetic form, but is transformed into 
 one or several other forms. If the amounts of each of these 
 be measured (by methods to be indicated), their sum total 
 will (under the principle of the conservation of energy) be 
 the total energy E transferred ; and thence F may be com- 
 puted. The descending clock weight is an instance of this 
 case. 
 
 Practical Measurement of Forces. — Forces may be meas- 
 ured by other phenomena which accompany them as well as 
 by the energy transferred. The resulting methods will of 
 course be secondary, that is, before being adopted they must 
 be shown to yield results consistent with the primary 
 (energy) method. 
 
Chap. VIII.] FORCE MEASUREMENT. 99 
 
 The most general of these consist in opposing to the un- 
 known force a known force just sufficient to balance it. 
 When no motion results, we know that the two forces are 
 equal and opposite, — a condition known as equilibrium of 
 the forces. This depends on the proposition that when 
 two equal and opposite forces are applied at the same point 
 of a free body, no motion ensues, — a proposition which 
 may be demonstrated as follows. If a body move under 
 the application of a given force, it will acquire energy at a 
 definite space-rate, and will therefore move over a definite 
 space in a given time, in the direction of the force. Suppose 
 such a force to be exerted for a short time and to be followed 
 by an equal and opposite force for an equal time, the body 
 will move a definite distance under the first force and an 
 equal distance in the opposite direction under the second 
 force ; so that, at the end of the second period, it will find 
 itself at the original starting-point. Suppose a succession 
 of such operations to occur; then the shorter the periods 
 of each, the more the action will resemble simultaneous 
 application of two equal and opposite forces. Also the ex- 
 cursion of the body acted upon will become smaller and 
 smaller, so that as the periods become shorter and shorter 
 we approach a condition of simultaneous forces and no 
 motion. 
 
 In the actual measurement of forces, or in an application 
 of known forces, use is made of this balancing of the effects 
 attending equal forces. The instruments and devices are 
 numerous, but they may be grouped for the most part as 
 applications of the principle of the lever, or of elasticity 
 (largely by use of the coiled spring). These methods imply 
 the ability to produce known forces (cf. p. 101), and in 
 this respect are to be distinguished from the energy method 
 of measuring force (p. 97). As typical of them the spring 
 balance will be briefly described. 
 
 Spring Balance. — If we suspend some free object from 
 
100 FUNDAMENTAL PnrSICAL CONCEPTS. [Part I. 
 
 the end of a coiled spring (taken merely as the type of all 
 elastic bodies), it will descend under the influence of its 
 weight, stretching the spring, and will finally come to rest 
 with the spring stretched so that its elastic force is just 
 equal and opposite to the weight of the body. The spring 
 will always show the same stretch (strain) when the same 
 object is hung upon it at the same point of the earth's 
 surface. But we know that under this condition, the 
 weight of the object is always the same (cf. p. 102). Hence 
 whenever we see the spring stretched to this amount and 
 held in equilibrium, we know that a force equal to that of 
 the spring, and therefore to the weight of the above object, 
 is exerted against the spring. It is of course assumed that 
 minor disturbing causes are eliminated. By attaching an 
 index to the spring and applying sundry weights, we may 
 graduate the instrument and thus have a spring-balance or 
 spring-dynamometer. Similarly a continued distortion of 
 any elastic body indicates not only a force in the body but 
 one against it. If motion is takiiig place, the two forces 
 are unequal when the state of motion is changing, equal 
 when the motion is uniform. 
 
 This operation of graduating the spring-balance implies 
 the ability to construct previously a graded set of bodies 
 which shall have known relative weights. The explanation 
 of how this is possible involves a most important practical 
 proposition in the system of physical measurement, and 
 this will be detailed in the following section. 
 
 Standards of Force. — As already stated, it is requisite, in 
 certain methods of measuring force and in the application 
 of known forces, to have at command the means of produ- 
 cing, at any time or place, accurately known forces whose 
 amounts are under control. For this purpose, we practically 
 employ the weights of certain bodies. These bodies consist 
 of suitably shaped pieces of metal or other durable sub- 
 stance, and are for convenience arranged in sets, each piece 
 
Chap. VIII.J FORCE MEASUREMENT. 101 
 
 having a weight which is a multiple or sub-multiple of the 
 weight of a standard piece of metal which is carefully pre- 
 served. The weight of this standard piece of metal, at a 
 standard locality, affords us our practical standard for the 
 measurement of weight, and of force in general. The two 
 recognized standard bodies are (p. 63) the "International 
 Kilogramme" and the "Imperial Pound Avoirdupois." It 
 will be shown later (p. Ill) how this graded set of bodies, 
 whose weight furnishes us with a means of determining 
 relative weights, also serves, through the assumed relation 
 between mass and weight, and through the definition of 
 weightal (p. 10), as a graded set of masses or weights for use 
 in the measurement of mass or weightal. The use of this 
 set of bodies in the primary measurement of kinergety has 
 been described in Chapter VI. It may here be again noted 
 that the standard body which serves to provide the standard 
 of force also serves as the standard of kinergety, of mass, 
 and of weightal. 
 
 The employment of the weight of a standard body as a 
 standard of force (not necessarily as a unit of force), de- 
 pends upon several considerations, namely : — 
 
 I. The constant intensity of weight (p. 82) at any given 
 point of the earth's surface. By " intensity of weight " is 
 meant the weight of the standard body. 
 
 II. The uniformity of weight throughout a space large 
 enough for experimental purposes (e.g. throughout an ordi- 
 nary room). 
 
 III. An accurate knowledge of the relative intensity of 
 weight at desired points upon the earth's surface. 
 
 All three of these points are capable of investigation by 
 two methods directly or indirectly. These are, the rate of 
 acceleration of a freely falling body, and the period of vibra^ 
 tion of an "invariable" pendulum. The law of universal 
 gravitation, and the constancy of this force between mem- 
 bers of the solar system, affords us corroborative evidence 
 
102 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 of the highest value as to the constancy of terrestrial 
 gravitation, except as this may be modified by local varia- 
 tions. 
 
 The constancy and uniformity of weight, or more properly 
 of the intensity of weight, at a given point and throughout 
 a given space might be deduced from the observed rate of 
 acceleration of a given freely falling body experimented 
 upon at different epochs and at different portions of the 
 space under consideration. This direct method, however, is 
 deficient in accuracy. 
 
 Observations with the " invariable " pendulum are those 
 upon which* we chiefly rely. But all such direct measure- 
 ments are of course confirmed by the absence of those con- 
 tradictions which would almost certainly arise in an immense 
 multitude of ordinary observations and phenomena if weight 
 were not constant at a given point. 
 
 The invariable pendulum consists of a metallic pendulum 
 so constructed that its kinergety (and therefore mass) and 
 its dimensions remain constant, and that its " length " (i.e. the 
 length of an equivalent simple pendulum) is unalterable 
 except through temperature, and so on. Such a pendulum 
 when swinging on knife-edges at a given point of the earth's 
 surface, has a time or period of vibration which depends 
 upon the kinergety and dimensions of the pendulum, and 
 upon the intensity of weight at that point. But as the 
 kinergety and dimensions are invariable, the time varies 
 only with the intensity of weight, and is inversely propor- 
 tional to the square root of that force. The intensity of 
 weight is proportional to the rate of acceleration (usually 
 denoted by g) which it will produce upon a body falling 
 freely in vacuum. The results of pendulum observations 
 are, for convenience, usually expressed by giving the value 
 of g, deduced from them for the place of observation, or the 
 relative value of g at that place as referred to that at some 
 standard point. 
 
Chap. VIII.] FORCE MEASUREMENT. 103 
 
 Through observations with such an invariable pendulum 
 or through other analogous pendulum researches, it has been 
 shown that the intensity of weight is constant from year to 
 year. If this were not so, clocks would require a continual 
 modification of the lengths of the pendulum to correspond 
 with the variations in weight, a fact which would become 
 very evident in astronomical observatories even if the varia- 
 tions were very small. Observations with invariable pendu- 
 lums at neighboring points show that the variations in the 
 same horizontal plane throughout a space such as an ordi- 
 nary room must be exceedingly small. Observations at many 
 and widely distributed points over the earth's surface have 
 given the relative value of g at these points with high accu- 
 racy, and from them the following general equation has 
 been deduced for expressing the relative value g of the 
 acceleration of weight at any point whose latitude is A 
 and whose altitude above the sea-level is H (expressed in 
 meters), 
 
 9 = 9*5,^0-- 0.00 269 cos 2 \ - 0.00 000 020 H), 
 
 9*6,0 = 980.6^^ approx. 
 sec' 
 
 This is the approximate average value of g at latitude 45°, 
 sear-level. Different experimental values of g^^^ o vary widely 
 in the next place of figures.' 
 
 From experimental observations with a simple pendulum, 
 the absolute value of g at any place may be determined with 
 an accuracy of about one part in one hundred thousand. 
 The relative determination at two different places by means 
 of the invariable pendulum may be made with at least ten 
 times this accuracy. The value computed for any particu- 
 lar place from the above formula would be liable to an 
 error exceeding one part in one hundred thousand. These 
 statements are intended to be only roughly approximate. 
 
 ' Very recent observations render it probable tliat near the earth's sur- 
 face the coefficient ol H is more nearly 0.00 000 030. 
 
104 FUNDAMENTAL PBTStCAL CONCEPTS. [Part I. 
 
 The total variation throughout the volume of an ordinary 
 room -would probably not exceed one part in one million. 
 The constancy of the weight of a body from one time to an- 
 other is undoubtedly far greater than the accuracy with 
 which it can be tested. 
 
 We may now proceed to show how a set of bodies which 
 shall furnish us with a graded set of weights may be con- 
 structed by the aid of the equal-arm balance. The accuracy 
 of the relative adjustment of these is limited only by the sen- 
 sitiveness of the balance, which is between one part in one 
 million and one part in ten million. Thus if we could use 
 this set of weights in connection with an equal-arm balance 
 we could obtain forces at any given point of the earth's sur- 
 face which would be relatively exact to the limit just stated. 
 If, however, we make use of this set of bodies at any other 
 point, they afford us forces whose absolute amount is differ- 
 ent according to the value of g at the place. This value may 
 vary by as much as one part in two hundred at extreme 
 points of the earth's surface, but between any places where 
 such work is likely to be done, the extreme variation would 
 not exceed one part in four hundred. Thus to assume that 
 the weights of this set are constant at all points of the 
 earth's surface is to introduce a possible error of one part 
 in two hundred ; while applying a correction for the relative 
 value of g will enable us to employ them at any desired 
 place with an accuracy of about one part in a hundred 
 thousand if the above formula is used, or with a higher 
 accuracy if a direct relative measurement of g is made with 
 an invariable pendulum. In using this set of bodies at 
 different places, therefore, for the production of known 
 forces, we must know the relative value of g at the place 
 where the weights are standard and at the place where they 
 are used, with the desired degree of accuracy. 
 
 The special application of the equal-arm balance in the 
 construction of the bodies having the graded set of weights 
 
Chap. VIII,] FORCE MEASUREMENT. 105 
 
 is as follows. It is the process called " weighing by taring " 
 and introduces no principle not thus far discussed in this 
 book. Suppose that we place in one pan of the equal-arm 
 balance the standard body, and produce equilibrium by 
 placing in the other pan sand or any convenient material 
 (a preliminary set of weights for instance). We thus evi- 
 dently exert through the beam of the balance at the knife- 
 edge carrying the standard body, an upward force equal to 
 the weight of that body. If now without disturbing the 
 counterpoising material, we should replace the standard 
 body by any other object, for instance the body whose 
 weight we desire to make equal to that of the standard, 
 there will not be equilibrium unless the weight of this 
 object is exactly equal to that of the standard body. We 
 may then add to, or take away from, the material of the 
 object until, after repeated trials, it is found that equilib- 
 rium occurs. The weight of the object is then precisely 
 equal to that of the standard body (within one part in one 
 million or less with suitable precautions and corrections). 
 
 Having thus primarily established two bodies having 
 equal weights, we may make a third which shall have a 
 weight equal to the sum of the two, and proceeding thus 
 we may build up a set of bodies whose weights shall be 
 in any desired gradation. This may be called a graded 
 " set of weights," although this term must be understood as 
 qualiiied by the variation of weight with locality. Assum- 
 ing for reasons stated in Chapter XI. that mass is propor- 
 tional to weight, this graded set of bodies may be and 
 generally is called a set of masses. It would be better 
 called a graded set of weightals (p. 153). 
 
 Unit of Force. — In choosing a unit of any kind whatever, 
 we are governed solely by considerations of convenience. 
 In framing a system of units for the measurement of quan- 
 tities of all sorts, it is of great advantage that the selection 
 should be systematic and not entirely arbitrary. By a suit- 
 
106 FUNDAMENTAL PHYSWAL CONCEPTS. [Part I. 
 
 able selection, it is practicable to avoid numerous incon- 
 venient mathematical constants in fundamental equations 
 connecting quantities of various kinds. This will now be 
 illustrated in the case of the unit of force. 
 
 Since force is measured by the space-rate of transference 
 of energy, the unit force will be such a force that under its 
 application, energy will be transferred at a unit space-rate. 
 If the force is constant, then under it a unit quantity of 
 energy will be transferred, when its point of application is 
 displaced through unit distance in the direction of its 
 action. Either of these statements constitutes a general 
 definition of the unit of force. We have, then, by the 
 proper choice of units 
 
 E = FS 
 
 without the introduction of numerical constants. 
 
 Next, inasmuch as the expression ^ MV^ has been uni- 
 versally accepted as the expression for kinetic energy, we 
 must have 
 
 where F equals the force (average or uniform) which, mov- 
 ing a free body of kinergety M, through a distance S, from 
 rest, produces in it a velocity V. 
 
 If a uniform force F acts upon this free body, the motion 
 will be uniformly accelerated. Let A represent the rate 
 of acceleration produced; then by the law of uniformly 
 accelerated motion, the velocity V, after the force has been 
 exerted through a distance S, will be such that, 
 
 V^ = 2 AS. 
 
 Substituting this in the above expression we have, 
 
 FS = iM.2AS, orF = MA. 
 
 The force F is therefore A times the number, M, of units of 
 kinergety in the body. This expression is general for all 
 forces, A being the observed rate of acceleration produced 
 
Chap. VIII.] FORCE MEASUREMENT. 107 
 
 on any kinergety M. The force under which a unit acceler- 
 ation would be produced on a free unit of kinergety would 
 then be a unit of force, since if A = 1, and M = 1, F will 
 be equal to one. We may, therefore, again define the unit 
 of force as that uniform force which will produce a unit rate 
 of acceleration in a unit kinergety. This is also a general 
 definition, and is a more convenient one to employ in con- 
 necting the unit of force with the standard of force. 
 
 Let us now apply it to this purpose. We know that any 
 kinergety, and therefore the i;nit of kinergety, falling freely 
 under its own weight at a point where the acceleration due 
 to weight is g (see p. 103), will be accelerated at that rate g. 
 Therefore, as the force is proportional to the rate of accel- 
 eration, the weight of a unit of kinergety must be g times 
 the unit of force. The weight of one gramme, at any place, 
 must, therefore, be g units of force, or in general : — 
 
 The unit of force, in any system, must be - of the weight 
 
 of a unit kinergety, where g is the acceleration due to 
 weight at the particular place. 
 
 We have now to connect this definition with a specific 
 system of units and with the standard of force. 
 
 The system of units in general use for scientific work 
 is called the " Centimeter-gramme-second System " (abbre- 
 viated c.g.s.). In establishing this system the units of 
 length, kinergety, and time are first selected, and are re- 
 spectively the centimeter, gramme, and second. The unit 
 of velocity then becomes one centimeter per second. This 
 unit of velocity has no specific name. The unit of force 
 which is called the dyne is the force under which energy 
 would be transferred at a space-rate of one unit (one erg) 
 per centimeter, or under which a kinergety of one gramme 
 would receive an acceleration of one centimeter per second. 
 By the above argument, the weight of one gramme at a place 
 where the acceleration of weight is g, would be g dynes. 
 
108 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 In other words, the unit of force, the dyne, is one g'th part 
 of the weight of one gramme. The value of g at the sear 
 level, in latitude 45°, is about 980.6 centimeters per second. 
 For other localities, if not known more exactly from pendu- 
 lum observations, it may be computed from the formula on 
 page 103. As the gramme is the thousandth part of the 
 kilogramme, we have thus connected the c.g.s. unit of 
 force with the weight of the kilogramme, that is, with the 
 standard of force. The unit quantity of energy is the erg, 
 which is the amount of energy transferred under a force of 
 one dyne, when its point of application is displaced through 
 a centimeter in the direction of the action.' 
 
 It is essential to clearness to remember always that 
 the words gramme, kilogramme, etc., in the c.g.s. sys- 
 tem, denote a kinergety, weightal, or mass and not a 
 force. Whenever force, or the particular force called 
 weight, are referred to quantitatively, the term dyne is 
 used. 
 
 It appears unfortunate that the unit of energy was not 
 originally defined in advance of the unit of force. If the 
 unit of kinetic energy had been defined as the kinetic 
 energy of a unit kinergety when moving with a unit veloc- 
 ity, the expression for kinetic energy would have been MV- 
 instead of \ MV^, which is now universally employed. We 
 should then have deduced the expression for force from the 
 equation FS = MV^, and should have arrived at the result 
 F = 2 MA instead of F = MA. The weight of the unit kin- 
 ergety would then have been 2 g units of force, and the dyne 
 would have been i as great as at present. The introduction 
 of the fraction i in the expression for kinetic energy, was 
 rendered necessary by the fact that the unit of force 
 was defined by the expression F = MA in advance of the 
 definition of the unit of kinetic energy. The expression 
 
 iFor a discussion of British and metric engineering units, see " Apple- 
 ton's Scliool Physics. " 
 
Chap. VIII.] FORCE MEASUREMENT. 109 
 
 ^ MV- is now so firmly established by custom, however, 
 that it will doubtless be permanent. 
 
 Weight Proportional to Kinergety. — That the relative 
 weights of bodies at any given locality are directly propor- 
 tional to their relative kinergeties is not axiomatic, but must 
 be proved experimentally. It is most simply demonstrated 
 by the fact that any two bodies of whatever weight, start- 
 ing from rest, will fall towards the earth with equal ac- 
 celerations provided that the retardation of air and other 
 resistances are removed. The relation follows from this 
 observation by the following argument. The force under 
 which a body descends, in the experiment, is its weight. 
 
 The energy received by each body in descending through 
 the same vertical distance S, starting from rest, is its weight 
 multiplied into the distance S, but the kinetic energy of 
 the two bodies after freely traversing this space S will be 
 in direct proportion to their kinergeties, K„ Kj, since their 
 velocities are experimentally shown to be equal, but the 
 kinetic energy must be in each case equal to the product 
 WS for that body. That is, for the first body Ei=WiS 
 and for the second body E2=W2 S. Now S is, by supposi- 
 tion, the same in both cases ; so that, 
 
 El :E, :: Wi: W^. 
 
 But, as already stated, Ej : Eg : 
 Therefore, Wi : W^ : 
 
 
 that is, the weights are in direct proportion to the kinerge- 
 ties, which was to be proved. 
 
 This experiment cannot be carried out with the requisite 
 accuracy, but its principle can be indirectly applied by 
 processes far more delicate. Of these the experimental 
 process on which we chiefly rely is the time of swing of 
 pendulums. Observations may be made with pendulums of 
 equal length consisting of heavy masses suspended by a 
 
110 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 fine wire. The periods of swing of such pendulums will be 
 found the same, independent of great diversity of material 
 employed in the construction of the separate pendulums, 
 and consequently in spite of great diversity of kinergety of 
 the pendulum. Newton devised and executed the first 
 experimental proof of this character. A more satisfactory, 
 because more accurate, proof comes from the observed time 
 of vibration of pendulums of different forms, where the 
 relative time of vibration is calculated from the form and 
 kinergety of the pendulum, under the assumption that 
 kinergety is proportional to weight. No contradiction be- 
 tween computed and observed times of such pendulums is 
 found, exceeding the limits of error of experimenting. A 
 further confirmation, and a still more general one, arises from 
 terrestrial and celestial mechanics. The elliptical forms of 
 the orbits of the planets, the relation between their distances 
 from the sun and their periods of revolution about it, and 
 their mutual influences upon each other and their satellites, 
 are all strictly consistent with the Newtonian law of gravi- 
 tation which asserts that the force of gravitation is propor- 
 tional to the masses ' of the bodies. Of this force, terrestrial 
 gravitation or weight is a special case ; moreover if the force 
 of gravitation were not proportional to kinergety, then, in 
 every planet, stresses of great magnitude would be developed 
 by the action of gravitation towards the sun. Of this, evi- 
 dence would certainly be given by disruption or distortion 
 of these bodies. No trace of any such effect is perceptible. 
 Further, if gravitation were not proportional to kinergety, 
 free bodies of different kinergety at the earth's surface 
 would not retain their relative positions, and a set of phe- 
 nomena would arise different from anything which we now 
 observe. This last consideration gives direct evidence that 
 weight is proportional to kinergety, while the . confirmation 
 
 1 On the assumption elsewhere stated that mass is proportional to kin- 
 ergety. 
 
Chap. VIII.] FORCE MEASUREMENT. Ill 
 
 of Newton's general law applies to weight only through the 
 assumption that this law holds with equal truth (so far as 
 it relates to mass) for bodies near each other, a proposition 
 whose reliability we have no reason to doubt in the case of 
 bodies near the earth's surface. 
 
 By the direct pendulum experiments, the proportionality 
 of weight to kinergety is established with an accuracy of at 
 least one part in a huadred thousand, while the confirma- 
 tions of the Newtonian law give to the proportion a far 
 higher probability. 
 
 From the proposition just established, it follows that a 
 set of bodies whose weights at a given point are graded 
 multiples or sub-multiples (p. 101) of that of the standard 
 body affords us a set of graded kinergeties which are multi- 
 ples or sub-multiples of the kinergety of the standard body. 
 The relative weights of these bodies are reliable within the 
 limit of one part in one million or better (the limit of the 
 equal-arm balance), and the relative Mnergeties are with 
 little doubt equally reliable. Thus we may assume, with a 
 very high degree of probability, that measurements of rela- 
 tive kinergety by the equal-arm balance are reliable up to 
 the limit of accuracy of that instrument. 
 
 Whereas the actual weights of these graded bodies (as 
 expressed in dynes or other unit similarly defined) vary 
 widely according to locality, the kinergeties are absolutely 
 constant, the same at all times and at all places, except when 
 the amount of substance in the bodies is changed through 
 accidental injury. 
 
 Weightal. — The quantity of any substance measured off by 
 " weighing " with the equal-arm balance against a standard 
 " set of weights " will be the same at whatever locality 
 the process is performed, although the weight of the portion 
 will vary. When we have to do with this quantity rather 
 than with force or weight, it will be called a weightal. This 
 term will be further discussed in Chapter XI. (cf. p. 152). 
 
CHAPTER IX. 
 
 WORK. 
 
 Nature of Work. — Let the term source or stock of energy 
 be used to denote some portion of energy of any form, the 
 action of wliich we are considering. Its location may or 
 may not be known, and it may or may not be associated 
 with some known substance. Inference from observation 
 shows that when a source of energy is exerting against a 
 body the action called force, and when the body undergoes 
 displacement in the direction of the force, then the source 
 gives up energy to the body at a definite space-rate. The 
 direction of the force has been defined as being the direction 
 of the motion of a free body under the force. Whatever 
 the path of the body in any case, the displacement in a 
 given direction is merely the change of position measured 
 in that direction ; or in other words, it is the component in 
 that direction when the actual motion is resolved into two 
 components parallel and perpendicular respectively to that di- 
 rection. The point of application and the line of a force are 
 the point at which the force is applied and a line through 
 that point in the direction of the force. If then a body on 
 which a source of energy is exerting a constant force F, 
 undergoes without rotation a displacement S in the direction 
 of the force, it will receive from the source a quantity of 
 energy E equal to the product of that space-rate into the 
 displacement. But as the space-rate is the measure of and 
 is numerically equal to the force E, the energy will be 
 E = FS. This transfer is entirely independent of any other 
 actions which may be in progress at the same time, and is 
 
 112 
 
Chap. IX.] WORK. 113 
 
 wholly independent of the path, except that this must have 
 the specified component S. If the force is not uniform, it 
 may be dealt with by the method (p. 99) of subdivision into 
 small steps. No new principle is involved in such a case. 
 This process of transfer of energy by the displacement of 
 a body against which force is being exerted is one case of 
 the performance of ivork. The quantity of work done W 
 is equal to the quantity of energy given up by the source, 
 in this case, so that W=PS. 
 
 The work is here said to have been done hy the acting 
 energy, because the displacement is in the direction of its 
 action, and also because the source has expended some of 
 its energy. 
 
 If the displacement S of the point of application of the 
 force, in the foregoing instance, had been in the opposite 
 direction, then the source of energy exerting the force F 
 would have acquired an amount of energy E=FS, precisely 
 the same as that which it expended in the other case ; and 
 this added energy would have been in the form of which 
 this source consists. These statements are also direct in- 
 ferences from observation. The action in this case is also 
 called the performance of loork. The amount of work is 
 again W=FS, but is now said to be done against the 
 energy source under consideration. 
 
 From the statements just made, together with the princi- 
 ple of conservation of energy, the following propositions 
 may be deduced. Suppose several uniform forces to be 
 simultaneously exerted in different directions at a single 
 point of a free body at rest, in a manner to produce no rota- 
 tion. The body will move with uniform acceleration in a 
 straight line. The direction of motion will be such that at 
 the end of any given time tlie body's position will be the 
 same as though the several forces had been successively 
 exerted, each for this same time-interval. Let Fj, Fj, F3, 
 etc., denote the respective forces exerted by the correspond- 
 
114 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 ing stocks 1, 2, 3, etc., of energy, which may be of different 
 forms. Let S be the actual displacement of the point of 
 application along the path, and Si, So, S3, etc., the corre- 
 sponding displacements in the direction of the forces ; that 
 is let Si be the component of S in the direction of Fi, S2 the 
 component in the direction of Fj, and so on. Then the 
 amount of energy expended by or imparted to the stock 1 
 will be FjSi. It is then said that work has been performed 
 by or against stock 1 to the amount '\Vi=FiSi. Like state- 
 ments hold for the other forces. Energy is expended 
 by the stock, that is work is done by it, when there is 
 displacement in the direction of its force. The work is 
 then accounted positive. If there is displacement in the 
 direction opposite to that of a force, say Fj, the correspond- 
 ing stock gains in energy of its own form, and work is said 
 to be done against the stock in amount W2=F2S2, and the 
 work is accounted negative. By the principle of conserva- 
 tion, the quantity of energy given up in this operation must 
 be equal to the total quantity accumulated. But the change 
 in kinetic energy of the body moved, constitutes in this 
 case a part of the accumulated energy. Hence the kinetic 
 energy of the body must be equal to the difference between 
 the positive and negative work done by the several stocks 
 of energy. This is called the resultant work done upon the 
 body. We may then in other words say that the resultant 
 work done upon the body is the algebraic sum of the work 
 done by the several acting stocks of energy, and that this 
 appears in the change in kinetic energy of the body. This 
 change may be either positive or negative. This proposition 
 may readily be shown to be general, that is, applicable to all 
 cases of motion of a body, whatever its initial state of motion, 
 and whether the forces are constant or variable. If the 
 forces are variable, the demonstration proceeds by the method 
 indicated at page 113. The operation may be regarded 
 as taking place in a succession of equal steps, each so short 
 
Chap. IX.] WORK. 115 
 
 that the forces may with but slight error be assumed con- 
 stant. We may then consider what the operation tends to 
 become as these steps are taken shorter and shorter. In 
 this way, the divergence of the imaginary, step-by-step 
 operation from the true one may be eliminated, and the 
 nature of the latter may be discovered. This process is the 
 basis of the calculus method. The kinetic energy dealt 
 with in this case is exclusively that of translatory motion. 
 If the point of application of the forces be such that rota- 
 tion ensues, the changes in the amount of kinetic energy of 
 rotation must be taken into account. This is also true if the 
 forces are not all applied at the same point of the body. 
 The general case is where each force has any direction and 
 point of application whatever, and the body is in any mode 
 of motion. The work done under or against each force is 
 then the product of that force into the displacement of its 
 own point of application in the direction of the force ; and 
 the resultant change in kinetic energy will be the sum of 
 the changes in kinetic energy of translation and rotation. 
 
 As a further illustration, take the case of a free body 
 moving in any - direction with any velocity, and exposed 
 after passing a given point, to a set of continuous, simul- 
 taneous, varying forces so applied that no rotation occurs. 
 The path of the body is determinable by the simple rule 
 that the position of the body at the end of any given time- 
 interval would have been that at which it would have 
 arrived had it traversed in succession the paths which 
 would have been passed over under the separate occurrence 
 of its own motion and of each of the stated forces. This 
 rule is a direct result of observation, and has already been em- 
 ployed in less general form. The change in kinetic energy 
 of this body can be determined under the general rule given 
 above, as may also the various amounts of work done and 
 their algebraic signs. The motion of the body will be ac- 
 celerated or retarded according to whether the resultant 
 
116 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 work done under the forces, that is, the algebraic svim of 
 the separate quantities of work, is positive or negative. 
 
 If two simultaneous, constant forces having different di- 
 rections be applied to a free body at rest in a manner to 
 produce no rotation, the body will acquire uniformly acceler- 
 ated motion in a straight line. The direction of motion and 
 the space-rate at which kinetic energy will be acquired may 
 be determined as above shown. This rate will be constant, 
 and the actual motion will be therefore the same as though 
 the body were moving under a single constant force having 
 the direction of this motion and applied at the same point 
 as the actual forces. Thus so far as the change of motion 
 of the body is concerned, and also the work done, the two 
 forces might be replaced by a single one as just described. 
 Such a force is called a resultant or resultant force. If the 
 body is in motion at the outset, the resultant force is not 
 thereby affected but is found as before. If there are more 
 than two forces, the resultant is found by the same proced- 
 ure. If the forces are variable in direction and anagni- 
 tude, the resultant has a specific value for each point of the 
 path, determined by the above method. In all cases of 
 several simultaneous forces at one point of a body, the 
 motion and work may thus be computed precisely as if the 
 body were under a single force equal to the resultant, simi- 
 larly applied, and due to one stock of energy. Whether or 
 not some of the energy exerting the original forces is trans- 
 formed into some other form within the body, thus produ- 
 cing a stock which actually exerts a force equal to the 
 resultant (and perhaps simultaneously produces other 
 forces), is a point to be studied for each individual case. 
 In cases where elasticity is brought into action, this seems 
 to occur. Such a point of view is helpful in dealing with 
 stress in liquids and solids. 
 
 In the discussion thus far given, the term ivo7-k has been 
 applied to the following three processes : The production 
 
Chap. IX.] WOEK. 117 
 
 of kinetic energy in a free body under a force ; the main- 
 tenance of the motion of a body against a resistance by its 
 own kinetic energy ; the production or maintenance of 
 motion of a body by energy other than kinetic against 
 resistance. 
 
 So far then as we have yet proceeded, the significance 
 attached to the term work miglit be expressed by the 
 definition : — 
 
 Work is that action of energy by which it produces motion 
 in a free body, or produces or maintains the motion of a body 
 against resisting force. 
 
 As thus regarded, work is that process of transformation 
 or of transference of energy which is characterized by the 
 displacement of a body under or against force. This state- 
 ment might be framed into a definition which should be 
 more explicit than the other. Either of them covers the 
 original and still common scope of the term. But there 
 are other transformations of energy in referring to which 
 the word work is now familiarly used to indicate both the 
 process and the quantity of energy transformed, although 
 neither force, nor motion of bodies, is discoverable. Ex- 
 amples of such are, — the heating of a wire by an electric 
 current ; electromagnetic induction ; the production of elec- 
 tric currents by heat at the junction of dissimilar metals ; 
 the production of heat by radiant energy. Although some 
 theories indicate the presence of both force and motion 
 of a material substance in some or all such cases, their 
 classification as instances of work would not thereby be 
 rendered admissible under the above definition. No other 
 convenient name for these processes has been offered, nor 
 does a separate term seem desirable. The fundamental 
 feature of the nature of work, as shown above, is that of a 
 process of exchange of energy. If we retain for the term 
 this denotation alone, dropping the limitations as to motion 
 and force, much will be gained in breadth and usefulness, 
 
118 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 and notliing of definiteness lost. The final definition tlieu 
 becomes : — 
 
 WoKK IS Any Peocess of Teansfekence oe Teans- 
 
 FOEMATION OF EnEEGY. 
 
 The term is also used by ellipsis to denote quantity of 
 ■work, a parallel with the quantitative use of the terms 
 energy, force, eta. The word transference is emploj-ed in 
 the sense of " imparting " from one portion of substance to 
 an adjacent one, not in the sense of transmission through 
 space. Consistently with this definition, the performance 
 ofivork means the transference or transformaiion of energy. 
 
 "Work is thus clearly to be apprehended as a secondary 
 concept, — a process, an action of energy, as force is, but of a 
 different nature. The distinction between work and force has 
 already been sharply drawn at page 41. Although, as will 
 be shown, quantity of work is numerically identical with 
 quantity of energy transformed, still ivork is not energy, nor 
 is energy work, — a distinction whose neglect finds abundant 
 illustration in current technical if not scientific literature. 
 
 Work by and upon. — It is consistent with the view here 
 taken to use the phrases, work done by or against .energy, 
 under or against a force, upon or against (in the sense of 
 upon) a body. It is convenient, but not thoroughly good 
 usage, to say, the work done by a body ; meaning the work 
 done by some energy resident upon or transmitted by the 
 body. It is not permissible to say, work done by or under 
 the action of a force, since the work is done by or under the 
 action of energy. 
 
 Quantity of "Work. — From the nature and definition of 
 work, it is sufficiently evident that the quantity of energy 
 transformed or transferred in any given operation is identi- 
 cal with the quantity of work performed. By the use of 
 consistent units the two become numerically equal. There 
 are thus two ways in which work done may be meas- 
 
Chap. IX.] WOBK. 119 
 
 ured. The first, or energy method, consists in measuring 
 the quantity of energy transferred. We may measure 
 either the energy expended or that which is produced. 
 Thus if the whole work in a certain case consists in acceler- 
 ating the motion of a body, we may measure the kinergety 
 and the initial and final velocities of the body. Thence 
 may be computed the increase of kinetic energy, which will 
 be the work done. Again, if an electric current heats a 
 wire, or if radiant energy heats a suitable body, and if the 
 heat produced is measured and expressed in units of energy, 
 this quantity will be the work done or energy transformed. 
 When the operation involves several transformations, then 
 all the quantities of energy produced must be measured 
 (separately if necessary) and their sum taken. The second 
 method is usually employed when the forces and displace- 
 ments occurring can be measured. The mode of calculating 
 the work done under each force from the known amount of 
 the force and the displacement of its point of application, 
 has been shown with sufficient fulness for the present pur- 
 pose, in the first section of this chapter. The sum of all 
 the positive quantities of work thus found, together with any 
 kinetic energy expended, will be the total work done, pro- 
 vided that no other forms of energy have been engaged in 
 the positive side of the action. Conversely, the sum of all 
 the negative quantities of work thus reckoned, together 
 with any kinetic energy produced, will be the total work 
 done, providing that no other energy than that thus meas- 
 ured results from the operation. 
 
 Units of Work. — The scientific unit of work in the c.g.s. 
 system is the erg, — the same as the unit of energy. It is 
 the work done when the point of application of a constant 
 force of one dyne is displaced through one centimeter in 
 the direction of the force. Work may be expressed in any 
 other unit in which energy is expressed, e.g. foot-pounds, 
 calories, etc. 
 
1:^0 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 Equilibrium of Energy. — When several forces are simul- 
 taneously exerted at the same point of a body, and no 
 change in state of motion thereby ensues, there is said to 
 be equilibrium of the forces. Whatever the path or mode 
 of motion of a body, the application to it of any set of 
 forces in equilibrium in no way affects that motion. In 
 such a statement, the space-rate which measures each force 
 is understood to be taken with respect to the same point of 
 reference as is the path or motion of the body itself. But 
 if the body is in motion, the common point of application 
 of the group of forces will undergo the same motion, and 
 work will therefore be done under or against each force as 
 already shown. Hence equilibrium of forces is not a suffi- 
 cient condition for equilibrium of energy, if by the latter 
 phrase be understood no performance of work under force. 
 Two conditions are necessary for no work under force; 
 namelj', equilibrium of forces, and no displacement (absence 
 of motion). Thus if a body at rest be under balanced forces, 
 no motion will ensue and no work will be done. If, now, 
 motion be imparted to the body by the brief application of 
 some other unbalanced force, then the motion will indefi- 
 nitely continue constant in velocity and direction so long as 
 the forces remain balanced, and work will be continuously 
 performed under certain of the forces and against others as 
 above shown. 
 
 Examples of Work. — An object thrown vipwards from the 
 earth performs work (or more properly its kinetic energy 
 does so) against its weight. The amount of work done in 
 rising through a vertical distance, whatever be the path, is 
 WH, where W is the weight of the object. The body would 
 rise to a height such that WH plus any other quantities of 
 work done (as against the resistance of the air) was equal 
 to the vertical component of the kinetic energy with which 
 the body started. This component is computed from the 
 vertical component of the initial velocity. The horizontal 
 
CuAp. IX.] WOBK. 121 
 
 component of the object's velocity is unaffected by the 
 action of gravitation, because at right angles to it ; and is 
 only lessened by the horizontal component of the air resist- 
 ance, etc. 
 
 There are many familiar cases of the continuous perform- 
 ance of work through approximately uniform motion under 
 balanced forces. Indeed the operation of most machines 
 affords illustrations. A railway train at full speed is an 
 example. As the train starts, the traction force is in excess 
 of the train-resistance (chiefly solid and air friction), and 
 the train therefore accumulates kinetic energy by increasing 
 velocity. The resistance increases as the speed increases, 
 so that with a level, straight track and a constant traction 
 force, a balance of forces will ultimately be reached. The 
 speed will then remain constant and work will be done at a 
 constant rate until some change of conditions occurs. 
 
 A pendulum clock driven by a " weight " is a somewhat 
 similar case. When the pendulum is at rest there is equi- 
 librium of forces, and the weight cannot start the pendulum. 
 But after the pendulum is set into a sufficiently wide swing 
 to free the escapement, an unbalanced force will be exerted 
 on it during a portion of each swing. Enough energy will 
 thus be imparted to it to maintain its motion against the 
 resistance of the air, etc. During each interval of freedom 
 of the escapement, the energy of the " weight " also ad- 
 vances the other mechanism through one step against its 
 frictional resistances. The transformations of energy in 
 more detail are as follows. 
 
 The case is one where a body moves against resistance 
 and where the acting energy is changed into another form, 
 and in this form is transmitted to the desired locality and 
 there again transformed in accomplishing the result sought. 
 The clock-weight receives gravitation energy at a definite 
 space-rate. The weight, as it descends, moves against the 
 resistance of the suspending cord. It fiist stretches the 
 
122 FUNDAMENTAL PUYSWAL CONCEPTS. [Fakt 1. 
 
 cord, thus transforming gravitation energy into energy of 
 elasticity in tlie cord until the elastic force in the cord 
 exactly equals the weight (assuming uniform motion). Af- 
 ter this, as the weight continues to descend, keeping the 
 clock in operation, the gravitation energy imparted to the 
 weight is transformed into energy of elasticity as fast as 
 the weight descends. This is transmitted from particle to 
 particle through the cord, through the drum on which the 
 cord is wound, and through the mechanism to the sundry 
 parts and surfaces where it is needed to maintain motion 
 against solid friction, the air friction, air resistance, etc. At 
 these parts, namely, bearings, gears, pendulum, and other 
 moving parts, the elastic energy is converted directly into 
 heat, or passes off in such other forms as energy of vibra- 
 tion (of which we have audible evidence), which in its turn 
 is reduced to heat. 
 
 Power. — When work is continuously performed, the time- 
 rate of the performance is called poiuer. Thus we may have 
 the definition : — 
 
 POWEK IS THE TiME-EATE OF THE PeRFORMAXCE OF 
 "WOKK, OB THE TiME-KATB OF THE TeANSFEEENCE OE 
 
 Teaxsfoematiou" of Exeegy. 
 
 If the work is being performed at a uniform rate, then 
 the power is constant, and the total quantity of energy 
 transformed is measured by the product of the power into 
 the time or duration of the performance. If the rate of 
 work is not uniform, then the average value of the power P 
 is W-hT, that is, the total amount of work W divided by 
 the time T occupied in its performance. Also the rate 
 of work at any time is the limit of this ratio W-t-T as the 
 value of T is taken smaller and smaller approaching zero 
 as a limit. The c.g.s. unit of power is one erg per second. 
 
 It is important to discriminate carefully between the use 
 of the terms power and energy. 
 
Chap. IX.] WORK. 123 
 
 The term power in technical usage has three distinct 
 meanings. The power of machinery is a term which prop- 
 erly denotes the rate at which the machine performs work, 
 and this is ordinarily expressed in the unit called the horse- 
 power, a unit based originally upon a rough approximation 
 to the rate at which a horse is capable of performing work. 
 Power is also used to denote a force applied to a machine, 
 as, for instance, in connection with the lever ; and it is also 
 used to denote a class of machines called mechanical powers. 
 
 Strength. — The term strength is familiarly employed 
 with a rather indefinite significance. It is used, however, 
 more nearly in the sense of force than of energy or power. 
 That is, we speak of a man who is capable of exerting a great 
 force, such as a powerful push or pull, as being strong, — 
 regardless of whether he may or may not be possessed of 
 endurance. A man would, on the other hand, rarely be 
 referred to as possessing great energy without implying 
 endurance. 
 
 Muscular Work. — This is merely the performance of 
 work by muscular energy already referred to at page 22. 
 A certain statement respecting muscular work is often 
 presented as a paradox, although really it is not one. It 
 appears to be true merely because one energy transforma- 
 tion involved in the case is overlooked. 
 
 If a heavy object is raised by the hand, work is done upon 
 the object in amount proportional to its weight and to the 
 height through which it is raised. This is done at the 
 expense of muscular energy and is, therefore, muscular 
 work. If, however, the object be held in the hand at a 
 constant height, no work is being done against gravity, that 
 is, no muscular energy is being transformed into gravitation 
 energy ; or, in other words, there is no displacement against 
 the resistance of gravity. It is, therefore, frequently as- 
 serted that no muscular work is being done. This statement 
 is a fallacy. Muscular work is being done in maintaining 
 
124 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 the muscles of the arm in the tense condition necessary to 
 support the weight. It is true that no work is being done 
 upon tlie object, but it is not true tliat no work is being 
 done in the muscles. A transformation of energy, resulting 
 in heat, goes on continuously in the muscles, corresponding 
 to the effort required to maintain their contraction. The 
 assertion in italics is, therefore, simply an inference, which 
 is erroneous because it overlooks this fact. 
 
 Methods and Units of Energy Measurement. — The methods 
 by which quantities of energy are measured, and the units in 
 which they are expressed, have been more or less indicated 
 in describing the several forms ; for the sake of complete- 
 ness, however, it may be worth while to summarize them here. 
 
 Kinetic energy, as shown at page 106, is measured by the 
 expression \ INIV^, the factors M and V being separately 
 measured quantities. In cases where it is not convenient 
 to measure 31 and V, the kinetic energy may be transformed 
 in any convenient way and the resulting quantity of energy 
 measured. Thus, for example, if the body be moving up- 
 ward freely, the height of its ascent and the amount of its 
 weight would, with a correction for air resistance, give a 
 measure of its kinetic energy, or, if that energy could be 
 wholly expended in the production of heat by the stirring 
 of water, or otherwise, the amount of heat produced would 
 measure the kinetic energy. 
 
 Those forms of energy which exert a force which can 
 readily be measured, are measured by the amount of work 
 done under or against this force. That is, the amount of 
 the force is measured as one factor, and the space through 
 which the point of application of the force is displaced in 
 the direction of the force is measured as the other factor. 
 The product of the two gives the work done and the energy 
 expended or received. If the force is not constant, then 
 the average force must be measured, or the expression con- 
 necting the force with the path of the point of application 
 
Chap. IX.] WORK. 125 
 
 of the force must be known, and the work done must be 
 calculated by means of this. Thus gravitation energy, the 
 energy of electrical and magnetic actions, elasticity, and so 
 on, are measured through the work done under their forces. 
 
 The energy of electrical currents is measured through the 
 product CVT, or its equivalent, where these letters denote 
 the current, the potential, and the time respectively. For 
 variable currents, the quantities C and V are the instanta- 
 neous values of the current and the potential ; and the total 
 energy-production or expenditure during the time T, must 
 be found either by the mathematical law of the change of 
 C and V with the time, or by employing an instrument in 
 which a mechanical action displaces the necessity for the 
 knowledge of this mathematical relation. Again, electrical 
 energy may be measured by the mechanical work which it 
 performs (with corrections for incidental losses). Or again, 
 electrical energy may be wholly transformed into heat and 
 measured as such. 
 
 Heat energy is measured by the number of units of mass 
 of water raised one degree in temperature. This arbitrary 
 unit has no simple relation to the units of length, mass, and 
 time, as have the units of the preceding measurements, but 
 is adopted solely because of convenience in manipulation. 
 
 Chemical energy, for reasons stated on page 65, is meas- 
 ured by the amount of heat which it produces. 
 
 The choice of units in the measurement of quantities of 
 energy is of course determined, as in all other measurements, 
 by custom based largely upon supposed convenience. In 
 scientific work, the unit of energy as measured through 
 work done against or under force, is the erg. The unit 
 quantity of heat is the calorie, the amount of heat which 
 will raise one gramme of water from 15° to 16° Centigrade, 
 the temperature being measured by the hydrogen thermom- 
 eter. This is the equivalent of 4.190-10^ ergs [" Com- 
 putation Rules and Logarithms," p. 72, The Macmillan Co.j. 
 
CHAPTER X. 
 
 POTENTIAL ENERGY. 
 
 Definition of Potential Energy. — This term is used to clas- 
 sify all forms of energy whicli are not assumed to be kinetic. 
 In this sense it has been employed to advantage in mathe- 
 matical physics for a long time. 
 
 The term potential energy has, however, a further signifi- 
 cance in regard to which much diversity of opinion has al- 
 ready been manifested and still exists. The terms actual 
 and potential were introduced by Kankine in 1853. As ap- 
 plied to energy, they appear clearly to have been intended 
 to distinguish between energy which a body actually has in 
 possession as kinetic energy, and energy which it has not 
 thus in possession but which is so conditioned with respect 
 to the body as to be acquired by it when free motion of the 
 body is permitted. There was no intention on the part of 
 Rankine, who was an able exponent of the doctrine of con- 
 servation of energy, to deny that this potential energy was 
 in existence previous to its acquisition by the body in mov- 
 ing. He merely regarded the energy as potential to, or pos- 
 sible to, as distinguished from being in possession by the 
 bod}'. This view seems distinctly to locate the energy as 
 outside of the body, or in other words, as acting upon it 
 from without ; although this may not have been "Rankine's 
 intention. While in the case of gravitation energy this view 
 as to locality may be strongly defended, yet in the case of a 
 bent spring the energy of elasticity, which is equally with 
 gravitation a kind of potential energy, appears almost cer- 
 tainly to be located upon the spring, and very possibly to 
 
 126 
 
Chap. X.] POTENTIAL ENERGY. 127 
 
 be as intimately associated with tlie particles of the spring 
 as energy of motion is with the particles of a moving body. 
 Electrical and magnetic energy, as well as chemical energy, 
 are naturally classed as potential forms ; and reflection will 
 show that while at first sight it may seem obvious that the 
 seat of the energy in these forms is within the individual 
 bodies concerned, yet more careful consideration will either 
 contradict this proposition or render it quite uncertain. At 
 best any decision as to the locality of any of these forms of 
 energy must result from a choice between hypotheses rather 
 than between known facts. Even in the case of heat, which 
 is not classed as potential energy, there is a great difference 
 in the order of certainty attaching to the assertions that heat 
 is a form of energy, and that heat is the kinetic energy of 
 random motion of the particles of bodies. The first of these 
 is as certain as anything in the doctrine of energy. The 
 second has thus far been found incapable of direct proof, 
 although the evidence in its favor is commonly regarded as 
 stronger than for the supposition that any other form of 
 energy is kinetic. ( Cf. Part II., Chap. V.) A fair-minded 
 and sufficiently pronounced position upon this subject, there- 
 fore, appears to lie in asserting merely that in the ease of all 
 the forms of energy which we choose to class as potential, 
 we do not know the locality of the energy. To assume on 
 the one hand that potential energy resides upon a body, in 
 the sense of being in full possession by the body concerned, 
 gives the mind at once a bias against any other view ; and 
 the same is true if the opposite view is chosen, namely, that 
 potential energy resides outside of, or is not in possession 
 by, the body. In the absence of logical evidence very 
 strongly in favor of one or the other of these views, it is 
 certainly most unscientific to adopt either. 
 
 The term potential energy has been used largely in a 
 restricted sense in dealing with mechanical energy, and it 
 was at first employed almost exclusively in this sense. 
 
128 FUNDAMENTAL PHYSICAL CONCEPTS. [Part 1. 
 
 Thus, in a system of bodies the total energy of which 
 depends both upon the configuration of the system and 
 on the motion of its parts, — that is, in whicli forces ex- 
 isted between the component bodies of the system, and 
 these bodies were changing position, — the total energy 
 was classified as of two kinds. The total energy due to 
 the configuration (or more properly, corresponding to it) 
 "was called potential energy ; the energy of motion was 
 called actual energy (by Kankine, 1853) or later kinetic 
 energy (by Thomson). Thus the term potential energy was 
 introduced by Rankine, " to denote that power of perform- 
 ing work which is due to configuration and not to activity." 
 (For the history of this subject, see "Philosophical Maga- 
 zine," xxxiii., p. 88.) 
 
 It is unfortunate that the phrase, " a body possesses po- 
 tential energy " and its equivalents are so deeply rooted in 
 scientific literature. It is probably impossible to displace 
 them, so that more than ordinary thoughtfulness needs to 
 be employed in their use. 
 
 Examples of Potential Energy. — The significance of this 
 term will be illustrated by a few examples. 
 
 When an object is held at some distance above the earth's 
 surface, we know that a force, its weight, is being exerted 
 upon it, tending to move it towards the earth. If it is 
 allowed to descend, it acquires eiiergy at a definite space- 
 rate. The amount of energy which it can acquire before 
 reaching the earth's surface is equal to WS, where W is 
 its weight and S is the vertical distance through which it 
 descends. At its elevated position, it gives no evidence of 
 the possession of this energy (as explained at p. 28), that 
 is, there is nothing whatever in the body to indicate greater 
 energy at its higher position ; on the contrary the force 
 upon it, if perceptibly different, is smaller than before. 
 We know, however, that in descending it will receive 
 energy in direct proportion to the vertical distance through 
 
CiiAP. X.] POTENTIAL SNSRGT. 129 
 
 which it moves. However many times the body be carried 
 back to its starting-point and again allowed to descend, it 
 will each time receive the same amount of energy. We are, 
 therefore, justified in saying that this quantity of energy is 
 so conditioned that it is inevitably imparted to the body in 
 its descent. This quantity of energy is, therefore, spoken 
 of as the potential energy of the body at a given point with 
 reference to the earth's surface ; that is, that quantity of 
 energy is potential, or possible, to the body in this position. 
 
 Similarly a bent spring, when released, acquires kinetic 
 energy, or is capable of imparting energy to other objects. 
 In its bent condition, therefore, a definite amount of energy 
 is potential to it, that is, it can and must receive this amount 
 of energy in coming back to its unstrained position. We 
 therefore say that the bent spring has or possesses poten- 
 tial energy, using the term with the same significance as 
 formerly in speaking of the potential energy of an object 
 under gravitation. The energy which is potential to the 
 bent spring, we call energy of elasticity. And it may be 
 worth while to repeat that, by classifying this as potential 
 energy, we do not intend to deny its existence at some 
 locality and in some form when the spring is in a bent 
 position, but merely to imply that it is not possessed by 
 the spring in the sense that a body in motion possesses its 
 kinetic energy, or at least that we do not know that it is so. 
 
 A piece of iron in the vicinity of a magnet, or of a coil of 
 wire conveying a current, is similarly said to possess poten- 
 tial energy ; two conductors conveying a current have poten- 
 tial energy with reference to each other ; two substances 
 which have " chemical affinity " for one another, possess 
 potential energy with reference to each other. 
 
 In the case of electrified bodies, the potential electrical 
 energy appears at first glance to reside upon the object. 
 An electrical charge, so called, is certainly present upon 
 them, and the presence of such a charge is a necessary 
 
130 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 condition of the energy ; but when we come to inquire 
 furtlier into the nature of the process, it becomes necessary 
 to assume some intervening medium by which the energy 
 exerts force upon the two bodies. To this medium is as- 
 cribed elasticity. The energy of the charges or charged 
 bodies may then possibly reside not wholly in the bodies, 
 but partly or wholly as elastic energy outside of them. This 
 illustrates the above made statement that the choice as to as- 
 cribing a locality to the energy is dependent upon hypotheses. 
 
 In all the above instances, and in all similar cases which 
 may be cited, energy is potential or possible to the body, 
 that is, the body has or possesses potential energy in virtue 
 jointly of its position with respect to some other body, and 
 of some store or stock of energy which is exerting force 
 upon one or both of the bodies, tending to make them ap- 
 proach or recede from one another. This is true either of 
 the bodj' as a whole, as in the case of gravitation, magnetic 
 and electrical action, and so on, or of the parts of a body 
 relatively to each other, as in the case of the bent spring. 
 
 In current phraseology potential energy is sometimes 
 called energy of position, since its amount depends upon the 
 relative position of the bodies or parts of the body. But 
 this name is misleading, since the position is merely one of 
 the elements in the case, — one of the factors of the energy. 
 The body would not possess the potential energy unless it 
 were in the given relative position ; but being in that posi- 
 tion, neither would it possess the potential energy unless 
 some store or stock of energy were exerting force upon it. 
 In current phraseology also, potential energy is said to be 
 due to the position of the bodi/ and to the action of the force 
 jointly. This statement, while true if associated with a cor- 
 rect definition of force, often misleads from the lack of such 
 a definition ; the production of the energy being then as- 
 cribed to the fo7ve instead of to the energy which produces 
 the force. 
 
Chap. X.] POTENTIAL ENERGY. 131 
 
 Any use of the term implying that potential energy is not 
 existent, or ascribing it merely to position, or ascribing it 
 to position and force without distinctly recognizing the re- 
 lation of force to energy, is productive of confusion. Such 
 use has been so common that the term has fallen into un- 
 deserved disrepute. The term is, however, unnecessary and 
 may be avoided without inconvenience, as has been done 
 throughout this book. 
 
 Quantity of Potential Energy. — Having in mind the rela- 
 tive character of motion, and therefore of force, we note 
 that when energy exerts a force, it does so upon each of at 
 least two bodies or portions of matter, tending to make them 
 approach or recede from one another, an equal force being 
 exerted upon each. In such a case, an equal amount of 
 energy (from the source producing the force) is potential to 
 each body ; or in the more common phraseology, each body 
 possesses an equal amount of potential energy relatively to 
 the other. If both bodies are free to move, each will receive 
 energy at the same space-rate, the force being equal upon 
 both ; and they will approach or recede with velocities (rela- 
 tive to their original points of rest or to their mutual center 
 of gravity) such that the accumulated kinetic energy of each 
 is proportional to the distance traversed from its original 
 point of rest. From this it is easy to deduce that the veloci- 
 ties will be inversely as the kinergeties ; or in other words, 
 that the momenta (kinergety x velocity) at the end of equal 
 intervals of time will be equal. 
 
 Thus, whenever a body is under the influence of a force it 
 possesses potential energy with reference to the other body 
 or bodies concerned in the action. The amount of potential 
 energy possessed by the body will depend directly upon the 
 amount of the force exerted upon it and the distance in the 
 direction of that force through which the body can travel. 
 The same form of statement holds, when, instead of dealing 
 with separate objects, we are concerned merely with parts of 
 
132 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 tlie same object. Thus a body above the surface of the earth 
 possesses potential energy with reference to the surface 
 jointly proportional to its weight W and to the vertical dis- 
 tance H above the surface. Its potential energy with ref- 
 erence to the surface is therefore ^YH. The earth possesses 
 the same xDotential energy with reference to the body. 
 
 A reference point for motion, force, and energy, fre- 
 quently employed is the so-called "center of mass" or 
 " center of gravity " of the system. It is not necessary 
 here to discuss the meaning or use of this term in any gen- 
 eral way. In the case of two bodies which are made to 
 approach one another by energy which is exerting a force 
 upon each of them, the position of the center of mass re- 
 mains unchanged by this action. This affords a reason for 
 its use as a reference point. 
 
 Suppose the motion of the body and the earth in the fore- 
 going illustration to be so referred. If then the body is 
 set free, the body and the earth mutually approach this 
 point with velocities inversely as their kinergeties, and 
 acquiring energies proportional to the respective spaces trav- 
 ersed. In view of the immense kinergety or mass of the 
 earth as compared with that of any ordinary body on its 
 surface, the velocity and kinetic energy acquired by the 
 earth in the time occupied by the body in its descent to the 
 surface is insignificant as compared with that of the body. 
 Thus in an ordinary case, the motion of the earth is entirely 
 insignificant and the earth may be considered as stationary 
 relatively to the center of mass of the system, and may be 
 used as the point of reference. 
 
 If the force is not a uniform one throughout the distance 
 that the bodies traverse in approaching one another, then 
 the potential energy is measured by the total work done 
 upon the body in traversing its path. If we let / denote 
 the mean value of the force while the body is traversing 
 any small distance s in the direction of the force, then the 
 
Chap. X.] POTENTIAL ENERGY. 133 
 
 work done upon the body as it traverses that distance is fs, 
 which is the amount of energy received by the body in trav- 
 ersing that space. The sum of all the successive portions 
 of work fs along the path from the beginning to the end 
 will be the total work done or the total energy received by 
 the body, and will therefore be the potential energy of the 
 body at its starting-point relatively to the final point of 
 the motion. If the force varies progressively, this summa- 
 tion will approach more and more nearly the true value of 
 the work, the smaller we make the successive intervals s. 
 Thus the total potential energy of a body at one point with 
 reference to a second point is measured by the total work 
 which will be done upon the body in passing from the first 
 point to the second point under the force concerned. This 
 work will be negative, that is, will be done hy the body in- 
 stead of upon it (p. 113), when, in order to move the body 
 from the first point to the second, it is necessary to move it 
 against the force considered. In that case, the potential 
 energy at the second position is greater than at the first. 
 
 In the case of a force tending to make two bodies ap- 
 proach one another, the potential energy of either relatively 
 to the other is greater the farther apart the bodies are situ- 
 ated, because obviously the amount of work which will be 
 done in bringing either body up to the other will increase 
 with their distance apart. The greatest possible potential 
 energy of one of these bodies with reference to the other 
 will correspond to an infinite distance between the two. 
 Since two bodies cannot be brought without change of form 
 nearer to each other than direct contact, the maximum 
 potential energy of one body with reference to another will 
 be measured by the work done in bringing either body from 
 an infinite distance up to contact with the other body. 
 
 Since the force tending to make two bodies approach or 
 recede from one another is in a straight line joining some 
 point of the two, and since the work done upon the moving 
 
134 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 body is only in proportion to its displacement in the direc- 
 tion of the force (that is to the component of its motion 
 parallel to the direction of the force), the total work done in 
 moving from one point to another, by whatever indirect path, 
 is the same in amount as by the most direct possible path, 
 namely, in a straight line joining the two bodies. Thus the 
 potential energy does not depend upon the path traversed 
 by the body, but only upon the displacement in the direction 
 of the force. 
 
 In the case of a force under which two bodies tend to 
 recede from one another, the considerations are precisely 
 the same as for bodies tending to approach one another, 
 except that the work is done in the opposite direction, and 
 that the maximum potential energy occurs when the bodies 
 are in the nearest possible position to each other instead of 
 in the most remote position. 
 
 The foregoing statements have dealt with potential energy 
 in cases where both or all bodies considered were acted upon 
 by the source of energy. But if we consider only one of 
 the bodies, and refer its position and motion to any arbi- 
 trarily chosen point, then the body will have potential 
 energy with reference to that point. This will be true, 
 except in the case where the reference point is located upon 
 an object which has always the same motion as the body 
 considered ; for, in that case, the potential energy of the 
 body, referred to the point, will be always zero. In the 
 general case, the body, when moving under the force, will 
 have its distance from the reference point either increased 
 or diminished. The total potential energy of the body 
 with reference to the point at any instant will be the work 
 which the body (or rather the energy moving the body) will 
 perform in traversing any path to the reference point from 
 its stated position. This work will be accounted negative, or 
 will be done upon rather than by the body, vi^hen the force 
 tends to make the body recede from the reference point. 
 
CHAPTER XI. 
 
 MATTER. 
 
 Distinction between Matter and Substance. — Aided by the 
 concept of energy, we may examine more closely that of 
 substance. This has been expressed as "that which is 
 assumed as existing in space, and as endued with powers 
 to aifect the human senses or portions of itself." But as 
 the powers have been shown to be the forms of energy, we 
 may describe substance as that which is assumed as exist- 
 ing in space, and as endued with its several forms of en- 
 ergy. We may remark first the hypothetical, or at best 
 inferential, character of the concept of substance. Obser- 
 vation shows us that different portions of space at a given 
 time contain different quantities of energy ; also that under 
 identical conditions different portions contain and can be 
 made to contain only unequally assorted forms and amounts 
 of energy. Into a given portion can be put, or with it can 
 be associated, a certain definite quantity only of each form. 
 The facts thus briefly alluded to suggest the idea of specific 
 capacities, that is, a definite capacity for each of the forms 
 of energy. But this idea is inconsistent with our notion of 
 space, and the thought therefore arises of something in the 
 space to serve as a receptacle or carrier of the energy, and 
 bearing these capacities as characteristics. Observation 
 further shows that the group of energy-forms may be trans- 
 ferred without change in a continuous path through space. 
 The forms composing such a group appear therefore to be 
 associated more positively than through mere position, as, 
 for instance, by being attached to something occupying that 
 
 135 
 
136 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 portion of space. Thus we arrive at the notion of a carrier 
 for energy, existing in space, and possessed of specific 
 capacities for energy of the several forms. Tlie capacities 
 are clearly an intrinsic part of the concept thus arrived at, 
 which is that of substance. The whole concept is clearly 
 supijosititious, or if one can find a siiitable major premise, it 
 may be called inferential. Its order of certainty, however, 
 is that which attaches to the existence of the human body, 
 or rather to the substance thereof. 
 
 Substance, then, whatever be its ultimate nature, must 
 be so organized as to be capable of having energy associated 
 with it in definite ways and amounts. Beyond this point 
 we have pure speculation alone. If more were known of 
 the nature of energy, some further advance might result; 
 but here, too, there is only hypothesis, although there are 
 some indications as to the manner in which some of the 
 forms of energy may be in association with substance. For 
 example, gravitation energy quite possibly acts on the 
 minute portions of substance from without; and likewise 
 cohesion energy. Heat, on the contrary, seems more prob- 
 ably to reside directly on these particles, perhaps as merely 
 the kinetic energy of their random motion. Elastic energy 
 seems to be dependent for its peculiar manifestation upon 
 some structural characteristic of the particles of substance. 
 Of electrical and rnagnetic energy, perhaps the less said the 
 better ; but the former appears external to, rather than as 
 acting within, the particles. Of chemical energy, one may 
 feel some confidence in assigning to structural functions of 
 the particles the selective capacities indicated by quantiva- 
 lence and combining weights ; but as to the seat and nat- 
 ure of the energy which produces and maintains chemical 
 union, and which manifests itself in the production of 
 heat, the mystery is, if possible, deeper. One may hazard 
 the thought, probably not novel, although no mention of it 
 comes to mind, that this energy is identical with that cans- 
 
Chap. XI.] MATTER. 137 
 
 ing gravitation and cohesion, a thought suggested by, and 
 not obviously inconsistent with, the vortex-atom theory of 
 matter (Part II., Chap. V.). 
 
 So far as the foregoing and like considerations can be 
 interpreted, they indicate no very simple character for 
 substance. Form of the particles of substance would nat- 
 urally first suggest itself as the basis of an explanation of 
 the intrinsic properties of substance. But it isTnot apparent 
 how mere differences in form could adequately account for 
 the chemical properties, or how form could afford an ulti- 
 mate explanation of elasticity. Moreover, form in the 
 physical universe is produced and maintained only through 
 the continued action of energy, — • usually of cohesion or 
 gravitation. Hence to assume form as the starting-point 
 would not go far towards simplifying our view of sub- 
 stance. No other simple suggestion offers itself. In the 
 entire line of speculations on the nature of substance, there 
 is but one which offers any real resolution of this concept 
 into simpler and well-recognized components. This is the 
 vortex-atom theory of which an outline is given in Part II. 
 
 The theory consists in the development of the proper- 
 ties which would be intrinsically possessed by a system of 
 vortex rings in a continuous " perfect fluid," and in com- 
 paring them with the known properties of substances. 
 The analogies thus found are most remarkable. The start- 
 ing-point is a hypothetical perfect fluid, continuous, and 
 uniform throughout space. In the plainest terms, this is 
 merely an imaginary primary substance, intrinsically de- 
 void of all energy, and so simple as to possess no prop- 
 erty but kiiiergety. Minute but multitudinous portions of 
 this fluid are assumed to be in motion as vortex rings. 
 These rings are the analogues of particles (atoms) of sub- 
 stance. In translatory motion, the rings possess kinetic 
 energy because of the kinergety of the fluid, but pass with- 
 out resistance through it. The vortex ring has elnsticity of 
 
138 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 form because of the energy of rotation of its constituent 
 fluid. Thus capacity for elastic energy is derived from a 
 mode of motion of this imaginary simple substance; and 
 elastic energy itself would be merely kinetic energy tempo- 
 rarily stored up in the distorted ring. The rings are not 
 only perfectly elastic but are also indestructible, add thus 
 afford all the basis necessary for the ordinary dynamical 
 theory of heat. Being capable of distinct periods of vibra- 
 tion they are adapted to meet the requirements of some 
 radiation phenomena. By their form and elasticity they 
 are peculiarly adapted to the bombardment theory of gravi- 
 tation and cohesion (possibly also chemical energy?). Fi- 
 nally, through certain structural differences (linkage, be- 
 knottedness, etc.) they acquire properties apparently identi- 
 cal with the chemical properties of quantivalence, atomic 
 "weight," and many others. All these points are developed 
 without further assumptions by purely mathematical pro- 
 cesses, and they can be disproved only by demonstration of 
 error in those processes, or by contradiction with fact. That 
 further developments of the theory, especially in relation to 
 electricity and heat, are lacking, is due perhaps more to the 
 immense mathematical difficulties attending the process 
 than to inadequacy of the hypothesis. This theory then 
 gives a definite concrete embodiment of the long-standing 
 speculation that the elements might prove to be resolvable 
 into a single, more elementary substance. To attain the 
 degree of simplicity naturally ascribed to it, this ultimate 
 substance must have no property dependent on energy or 
 even on motion. The vortex theory will meet even this 
 severe condition provided that the one property, kinergety, 
 of the hypothetical " fluid " can be shown not to arise from 
 energy. There is no reason for the supposition that it does 
 so arise, but on the other hand no strict disproof is forth- 
 coming. A discussion given in Part II., Chapter V., shows, 
 however, that mere continuous permanent occupancy of 
 
Chap. XL] MATTER. 139 
 
 space is an entirely adequate basis for kinergety, at least 
 so far as that property is involved in this theory. 
 
 We may then say that such evidence as we have, to- 
 gether with rational and detailed speculative theory, indi- 
 cates that substance is not simple in character, and gives 
 support to the suggestion of its resolvability into two com- 
 ponents. These are energy and an inert underlying sub- 
 stance devoid of all characteristics save kinergety or some 
 even more elementary property to which this may be due. 
 
 In view of the avowedly speculative character of our ac- 
 quaintance with the nature of substance, are we justified in 
 permitting it to exert any influence in the formation of our 
 fundamental physical concepts ? As to this, on the one 
 hand, we cannot too strongly insist that no such considera^ 
 tions as those above advanced afford any pi-oof whatever of 
 the resolvability of substance. On the other hand, to 
 ignore the possibilities thus opened up would be equally in- 
 judicious. To rest the case at this point might be the more 
 logical procedure, but the apparently fixed trend of the 
 mind to seek the simpler under the more complex has es- 
 tablished in physics the custom of assigning the name 
 matter to the inost elementary concept of that which under- 
 lies all bodies or substances and out of which they are made 
 up. So long as the doctrine of energy was in abeyance, the 
 concept of matter included not only that of substance as 
 here defined, but also all or nearly all of the "powers." 
 Now, however, matter must be so defined as to exclude the 
 concept of energy. But to accomplish this end as effectu- 
 ally as may be, regard must be had to all reasonable possi- 
 bilities as to the occurrence of energy as a constituent of 
 substance. It is precisely here that the speculations above 
 given are entitled to place. Not that the denotation of 
 matter should recognize or give expression to the hypoth- 
 eses underlying any theory, but that the definition should 
 if possible be so framed as to exclude no rational view of 
 
140 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 the nature of substance or matter. Should it be remarked 
 that so abstract a concept of matter is a metaphysical rather 
 than a physical one, it may be rejoined that the definitions 
 seek only to reduce the concepts of matter and of substance 
 to the lowest terms, not to analyze them. Such analysis 
 would lie beyond the domain of physics and would fall 
 within the scope of the modern psychology. 
 
 Definition of Matter. — If we employ the term inert as 
 meaning intrinsically devoid of energy and motion, the fol- 
 lowing appears to be a definition consistent with the condi- 
 tions of the problem ; — 
 
 Matter is the Inert Constituent of Substance. 
 
 There is here no implication that it is physically possible 
 to separate energy from matter. The concepts of energy 
 and matter are mutually exclusive, but this in no way re- 
 quires or implies or denies the possibility of separation in 
 space and time. 
 
 The remark may be again recorded that the concept of 
 matter would become identical with that of substance if the 
 capacities could be demonstrated not to be due to energy. 
 The capacities would then become intrinsic properties of 
 matter, and we should be obliged to recognize some seventy 
 forms or kinds of matter, whereas we now admit these ele- 
 ments only to the rank of kinds of substance. This pos- 
 sibility is not ignored in the above definition, but the 
 presumption seems at present much against such an out- 
 come. 
 
 Matter Non-resistant. — It is a common misapprehension 
 that a portion of matter, even when free, resists an effort to 
 set it in motion or to accelerate its motion. This mistake 
 arises from two combined causes, namely, the disregard of 
 the distinction between matter and substance, and the fact 
 that substances or bodies do exert such resistance, or rather 
 that the elastic energy called into play upon them does so. 
 
Chap. XL] MATTER. 141 
 
 When a body is pushed by the hand or by another body, we 
 know that it exerts an opposing force due to elasticity, 
 whether or not it be free. The distinction between matter 
 and energy shows that this force can be exerted by energy 
 only and not by the matter of the body. It is true that we 
 do not know any substance devoid of elasticity, that is, one 
 which is incapable of showing some of the phenomena of elas- 
 ticity under proper test ; whence it might be said that we are 
 not justified in assuming that matter can exist devoid of elas- 
 ticity and therefore incapable of exerting force. The fal- 
 lacy in this remark lies in the last few words. It is quite 
 possible that matter does not, perhaps can not, exist without 
 elasticity, but none the less it is the energy and not the 
 matter which exerts the force. Even granting what has 
 not been proved, that matter does not exist without elas- 
 ticity, the independence of the two concepts, matter and 
 energy, is in no way impaired thereby, and there is no rea- 
 son for confusing them hopelessly, as would be done by at- 
 tributing to matter the exclusive attribute of energy, — the 
 exertion of force. 
 
 When force is exerted against a body, the latter does not 
 acquire its full velocity instantly, but its speed increases 
 only at a definite rate. This is sometimes cited as indicat- 
 ing a " resistance " of matter to being moved, and also of 
 the " inertia " of matter. It indicates neither. By our defi- 
 nition of force, we recognize the fact that when a free 
 body moves under a force, it acquires kinetic energy at a 
 definite space-rate, no more no less, at every point of its 
 path, this space-rate being used as the measure of the force. 
 A freely moving body, therefore, can have its motion accel- 
 erated under a certain force at no greater rate than will 
 correspond to this space-rate of transference of energy to it. 
 The body will be accelerated at such a rate that, supposing 
 it to start from rest, its velocity at any point of the path 
 will be such that the kinetic energy of the body will be 
 
142 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 equal to the energy which has been imparted to it, namely, 
 the force multiplied by the distance through which the body 
 has traversed in the direction of the force. Since the kinetic 
 energy is proportional to the kinergety and to the square of 
 the velocity of the body, this velocity at any given point of 
 its path, or the time of acquiring any specified velocity, 
 will depend upon the kinergety of the body. A body with 
 greater kinergety, after having traversed under the same 
 force a given distance along its path, will possess the same 
 energy and a less velocity than a body of smaller kinergety. 
 
 If the matter of the body exerted an opposing force, that 
 is a resistance, kinetic energy would be acquired at a rate 
 proportional only to the difference between the two forces, 
 which is contrary to the fact. At first glance this last state- 
 ment may seem counter to the admission that an actual 
 body even if free does exert a resistance equal and oppo- 
 site to that exerted against it, as for instance in the colli- 
 sion of elastic bodies. But not only is this resistance, as 
 just shown, due to the elastic energy called into action upon 
 the body being moved, but also this energy is only the trans- 
 mitting agent between the acting source and the matter of 
 the body ; that is, the elastic energy exerts an unresisted 
 force on the matter of the body and in the direction of the 
 external force upon the body. To associate with the con- 
 cept of matter any idea of resistance to motion is not merely 
 flatly contradictory to the entire line of presentation adopted 
 in this book, but is irreconcilable with any consistent view 
 of the phenomena of physics. 
 
 Properties of Matter. — The many properties of substances 
 need not be here catalogued in full. To mention exten- 
 sion, inertia, elasticity, cohesion, gravitation, magnetization, 
 chemical affinity, electrical conductivity, etc., is sufficient 
 to indicate what is meant. If the foregoing distinction be- 
 tween matter and substance be disregarded, intentionally or 
 not, the terms become coextensive. This custom, however 
 
Chap. XI.] MATTER. 143 
 
 clearly reprehensible, has been and still continues to be 
 almost universal, as also has the corresponding omission 
 to discriminate between properties of matter and of sub- 
 stances. But as the ideas of matter and energy are mutually 
 exclusive, any property demonstrably due to the presence of 
 energy is therefore not a property of matter, but only of 
 substance. Further, no property of substances can be rig- 
 idly ascribed to matter, unless demonstrated not to arise 
 from energy. Bearing in mind the inferential character of 
 the properties which we have called capacities, our small 
 knowledge of their nature if they do exist, and also the 
 rational analogies to them based on the mere motion of a 
 hypothetical inert "matter" without energy except in virtue 
 of its motion, we must deny place as properties of matter 
 to all the properties of substances, at least pending proof, 
 except possibly to extension, inertia, and kinergety. These 
 must be further considered. 
 
 Extension. — As to extension, the volume of bodies is 
 determined in part at least by the quantity of heat energy 
 which they contain. What the volume of any body devoid 
 of all heat energy would be, is a matter of speculation ; and 
 whether such volume would be the volume of the matter 
 contained in the body, or even whether it would indicate 
 that this matter when devoid of energy, possessed volume, 
 cannot be affirmed. The property of extension, that is, the 
 occupation of space, is not deducible as a property of matter 
 from the fact that substances occupy space. It is in reality 
 only through speculative considerations of a sort to be 
 alluded to in Part II., that this property has been ascribed 
 to matter. We may, therefore, properly deny that it pos- 
 sesses a legitimate claim to such recognition. 
 
 Inertia. — The so-called property of inertia is in reality 
 not a property but a principle or doctrine (p. 19). Viewed 
 under the modern definitions of matter and energy, the prin- 
 ciple falls to a position of comparative insignificance. Any- 
 
144 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 thing which changes the state of motion of a body is, by 
 definition, energy. Matter cannot, therefore, produce such 
 a change in state of motion, and hence is inert. Thus the 
 principle is a direct outcome of the definitions. It must 
 however be noted that the concept of capacity for kinetic 
 energy (kinergety) is often associated with or merged into 
 tliat of inertia. Bodies are said to possess their kinetic 
 energy in virtue of tlieir inertia, and to " resist motion " also 
 because of that inertia. This practice results from a con- 
 fusion of ideas. Inertia and kinergety have no relation 
 whatever to one another. 
 
 Kinergety. — The property which we have called kiner- 
 gety appears to belong universally to all portions of substance 
 however minute, and to be inseparable from them. It ap- 
 pears in no way to depend on any amount or form of energy 
 in the substance, but of course we can only assert this prop- 
 osition as true within those limits inside of which we are 
 able to remove or add to these quantities of energy. If it 
 should be true, as some theories suggest, that substance can 
 exist as such only through the association of some form of 
 energy with some portion of matter, or through some mode 
 of motion of matter, then it is conceivable that the kinergety 
 of the substance might result from some other property in 
 the underlying matter, — such for instance as mere perma- 
 nent occupation of space. However much or little weight 
 we may assign to such hypotheses, as explanations of the 
 real nature of matter and substance, still, in view of the 
 possibility which they present, we must decline to admit that 
 the possession of kinergety by matter is a necessary inference 
 from the possession of this property by substances. It may 
 be well in view of the future use which is to be made of it, 
 to restate here some of the evidence upon which kinergety is 
 classed as a universal attribute of substance. 
 
 A body, moving with a definite velocity with reference to 
 a given point, possesses a definite amount of kinetic energy 
 
Chap. XI.] MATTER. 145 
 
 with reference to that point (p. 60). But at the same in- 
 stant it possesses with reference to any other point a definite 
 other quantity of kinetic energy, the difference being deter- 
 mined solely by the difference in velocity of motion of the 
 body relative to the two points. The kinergety of the body 
 is of course in no way affected by the mere change of refer- 
 ence point of the motion. Again, a body undergoing no 
 apparent change, e.g. a piece of metal, possesses at differ- 
 ent times always the same kinetic energy when its velocity 
 is the same. Again, if we add to or abstract from such a 
 body as the piece of metal just cited any amount of sub- 
 stance, we find that at the same velocity its kinetic energy 
 is respectively increased or diminished (within the limit of 
 sensitiveness of the methods employed). The converse of 
 this statement is also true. Again, the kinetic energy of a 
 body which undergoes no perceptible increase or diminution 
 of substance is the same at the same velocity, wha,tever the 
 external condition, and whatever the amounts of energy of 
 other forms which it may possess. These propositions are 
 established upon no simple experiments but are confirmed 
 by their consistence with universal experience in all branches 
 of physics. Beyond this we have the general principle of 
 the conservation of kinergety (p. 156). From all these, the 
 inference is unmistakable that kinergety is an intrinsic 
 property of all substances and of every minutest portion of 
 them. It appears also that the kinergety of any portion 
 of substance is a property wholly independent of the exist- 
 ence of any other substance or body, and of any other 
 property or form of energy. 
 
 Gravitation. — It may be well to consider somewhat fur- 
 ther such a property as the capacity for gravitation energy. 
 The actiort-of gravitation is clearly due to a form of energy. 
 The capacity for gravitation energy may be defined as being 
 the energy which a given body would acquire in moving 
 through a specified distance in the direction of a second 
 
146 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 body, the two being at a specified distance from one an- 
 other, and no other bodies being concerned in the action. 
 If the second body and the distances were to remain the 
 same, and the first body were replaced successively by 
 others, the quantity of energy received by each of these 
 several bodies would be in general different for each of the 
 bodies, and would be the same for any one of them at all 
 times. These quantities may be taken as the relative 
 capacities for gravitation energy of the respective bodies. 
 But it must, of course, be understood that these quantities 
 are a function not only of these bodies but of the second body 
 which remains constant in all of the supposed cases. These 
 capacities we know by JSTewton's law for universal gravitar 
 tion to be directly proportional to the kinergety of the 
 respective bodies. For although Newton's law as ordi- 
 narily expressed states them to be proportional to the 
 respective masses, the term mass as there used really 
 represents what we have called kinergety or capacity for 
 kinetic energy. The capacity may be otherwise defined, as 
 for example, by stating that the capacity of a system of two 
 bodies is the amount of energy acquired by them in moving 
 through a specified distance towards one another when at a 
 specified distance apart. As so defined the capacity is by 
 Newton's law proportional to the product of the kinergety 
 of the respective bodies. Or the capacity may be treated 
 from other points of view but with substantially the same 
 result. This capacity for gravitation energy is as intimately 
 associated with every minutest portion of substance as is 
 kinergety, and so far as known is also universally so asso- 
 ciated, but there is even better reason for refraining from 
 the inference that it is a property of matter than in the case 
 of kinergety. The plausibility of the hypothesis which 
 explains gravitation as due to the kinetic energy of a cer- 
 tain system of moving particles of substance, combined with 
 the argument just presented in the case of kinergety to the 
 
CuAr. XI.] MATTER. 147 
 
 effect that a property may attach to a portion of substance, 
 and still not attach to the inert matter composing that por- 
 tion, indicates clearly that matter itself may possess no 
 capacity for gravitation. 
 
 Mass : Quantity of Matter. — Whatever concept of matter 
 be adopted, the idea of quantity is associated with it. This 
 idea is denoted by the term viass, — a term which carries 
 with it no further significance whatever. Hence, 
 
 Mass is Quantity of Matter. 
 
 It is not a property of matter, neither does it refer to any 
 such property. Although often regarded as arising from or 
 related to inertia, it has absolutely no connection with that 
 principle. The term is nothing more than a verbal substi- 
 tute for the words quantity of matter. 
 
 Inasmuch as we are and can be acquainted with matter 
 only as manifested in substance, quantity of matter can be 
 measured only through some property of substance. In 
 other words we can measure only quantity of substance. 
 To give recognition to this fact by defining mass as quantity 
 of substance rather than of matter would be more logical. 
 But custom has fixed the latter usage, through the long 
 employment of matter, with the denotation more strictly 
 pertaining to substance, as above shown. This is perhaps 
 not a serious disadvantage ; for if substance differs from 
 matter, it does so through being matter organized through 
 motion or energy which is in permanent association with it. 
 In this substance thus organized into the possession of prop- 
 erties, it is often assumed that it is only with the inert com- 
 ponent that we are generally concerned when measure- 
 ments of quantity are to be made, that is with matter. If 
 substance bears to matter the relation supposed, then 
 a designated quantity of substance consists of a definite 
 quantity of matter in permanent association with a definite 
 quantity of energy or motion. Thus in whatever manner 
 
148 FUNDAMENTAL PHYSICAL CONCEPTS. [1'akt I. 
 
 quantity of substance be measured it will bear some definite 
 relation to quantity of matter. The nature of this relation 
 will, however, depend on the method used. Hence the 
 selection would naturally be made with a view to securing 
 the simplest possible relation, namely, that of direct propor- 
 tionality. If we had been able to demonstrate with cer- 
 tainty any one property of matter, this would without 
 question be made the basis of the definition of quantity of 
 matter, that is of mass, even if not adapted to the practical 
 measurement of mass. In default of this, a property of 
 substance must be chosen. The conditions to guide the 
 choice are : First, the property should be such as to render 
 highly probable the assumption that its adoption will yield 
 quantitative measurements directly proportional to mass ; 
 second, the property must be such as to lead directly or 
 indirectly to a practical method of measurement of demon- 
 strably sufficient accuracy. Let us now proceed to the 
 selection. 
 
 The several " powers " of substances, even if regarded as 
 properties of substance, are debarred from service, as the 
 amount of any of them associated with a given portion 
 of substance varies with time and place. The choice is 
 therefore restricted to the "capacities." Of these the 
 chemical properties are obviously unsuitable, as they are 
 not common in either kind or degree to all substances, and 
 can obviously bear no common or simple relation to mass. 
 The same is clearly true of the electrical, magnetic, elastic, 
 cohesive, and heat properties, as well as of properties re- 
 lated to radiation. The choice, thus narrowed, lies between 
 kinergety and capacity for gravitation energy, or some 
 manifestation of it (such as weight). 
 
 If a homogeneous body, that is, one of which every por- 
 tion is identical in properties with every other portion (so 
 far as can be discovered), be cut into parts of equal volume, 
 these parts would naturally be assumed to contain equal 
 
Chap. XL] MATTER. 149 
 
 quantities of matter, that is equal masses. This appears to 
 be the most primary view attainable of equal masses. But 
 by weighing (p. 104) these equal volumes will be found to 
 have equal weights and therefore (p. 109) equal kinergeties. 
 Hence there is strong presumption that equal masses of the 
 same substance have equal weights and equal kinergeties. 
 No such direct evidence exists as to the relation of the 
 weight or kinergety of a given mass of a substance of one 
 kind and that of an equal mass of another, for the obvious 
 reason that equal volumes of different substances probably 
 have masses which are not only unequal but stand in no 
 primarily determinable relation to each other. "We have 
 therefore to resort to . more indirect considerations, and 
 these are furnished by a review of the properties of kiner- 
 gety and gravitation. This, however, has already been 
 made (p. 114). We have seen that kinergety pertains to 
 every portion of substance ; that there is nothing to in- 
 dicate that it may not be an intrinsic property of matter as 
 well as of substance, or at least the direct result of an in- 
 trinsic property of matter, since it does not appear to re- 
 quire either energy or motion for its explanation ; that as a 
 concept it is essentially simpler than that of capacity for 
 gravitation energy ; that it is capable of primary meas- 
 urement with moderate accuracy, but of indirect measure- 
 ment with great accuracy by '■' weighing " (p. 104). As to 
 capacity for gravitation energy, ^\'e have found it less 
 simple in character although no less universal as an attri- 
 bute of substance. But all the facts concerning gravita- 
 tion would be entirely met by the supposition that this 
 capacity does not attach to matter but only to substance. 
 ISTeither is the capacity itself, as above defined, any more 
 directly measurable than kinergety. Weight as a manifes- 
 tation of gravitation might be suggested as a substitute for 
 the capacity for gravitation energy as a measure of mass. 
 This is based on the common assertion that weight is pro- 
 
150 FUNDAMENTAL PBTSWAL CONCEPTS. [Pakt I. 
 
 portional to mass. But for the reason just stated in consid- 
 ering the capacity itself, this is evidently pure assumption. 
 Moreover weight varies with locality, and therefore cannot 
 be proportional to mass, which is constant. 
 
 There is thus a distinct preponderance of reasons for the 
 selection of kinergety as the property through which to de- 
 fine quantity of substance and upon which to establish the 
 measurement of mass. This choice has also the sanction of 
 custom. We therefore assume that mass is directly propor- 
 tional to kinergety. Also as weight has been demonstrated 
 to be directly proportional to kinergety, we assume that mass 
 is directly proportional to weight — the weight being always 
 understood as taken at or corrected to a fixed point of the 
 earth's surface. The propositions just enunciated may be 
 stated in another form as follows : Tlie relative masses of 
 bodies are assumed to be to each other in the ratios of the 
 Mnergeties of the bodies, or of their weights at a given locality. 
 This statement brings out the point which is to be clearly 
 appreciated, that like all other physical quantities, mass 
 is purely relative. Masses are referred to a chosen stand- 
 ard of mass, the customary standards having already been 
 named. 
 
 The existence of a broad assumption is generally over- 
 looked in asserting the proportionality between mass, kin- 
 ergety, and weight. The assumption exists whatever form 
 we give to the concept of matter, and is not a result of this 
 particular mode of presentation. The oversight arises 
 from failure to recognize kinergety sufficiently or at all as 
 a distinct property of substance. The kinetic energy of a 
 body is ordinarily said to be proportional to its mass, with 
 no allusion to kinergety. Hence inevitably arises the mis- 
 taken idea that mass is a 2'^^'operty of substance or matter. 
 There is perhaps no more prevalent misconception in the 
 range of physics. 
 
 It is to be noted that we arrive at the same results 
 
Chap. XI.] MATTER. 151 
 
 whether relative weights or relative kinergeties be adopted 
 for the measurement of mass, because of the proved propor- 
 tionality between weight at a fixed point and kinergety. 
 But it is also to be borne in mind that weight is merely 
 a force whose intensity (cf. p. 101) varies with locality, 
 so that bodies of equal mass or kinergety do not have 
 equal weights at different places on the earth. Equal 
 weights at different geographical localities do not therefore 
 measure off equal quantities of matter or of substance. 
 
 As presumptive evidence of the correctness of the assump- 
 tion of direct proportionality between mass and kinergety, 
 may be cited the comparative simplicity of the laws of 
 matter and energy framed on that basis. Complexity 
 might naturally be assumed as the sure result of an error 
 in this fundamental assumption, although such a supposi- 
 tion is not capable of proof. 
 
 Practical Measurement of Mass. — All physical phenom- 
 ena arise from the action of energy associated with sub- 
 stance. Hence most physical and chemical measurements 
 have to do with quantities of energy associated with quan- 
 tities of substance or matter. A sufficiently delicate prac- 
 tical means of measuring quantity of substance or quantity 
 of matter is thus a necessity. The usefulness of such a 
 process is enhanced by the fact that under given conditions 
 of temperature, etc., a stated quantity of a given substance 
 bears always the same quantities and forms of energy ; so 
 that the measurement determines incidentally a precise 
 quantity of energy. The practical measurement is a process 
 which enters so fundamentally into every kind of quanti- 
 tative work that higher delicacy is demanded in it than in 
 any other. Fortunately this is attainable without undue 
 labor by the use of the equal-arm balance (p. 104) in the 
 process called "weighing." The limit of accuracy is from 
 one part in one million to one part in ten million (p. 105). 
 The process has already been sufficiently described. 
 
152 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 "Weightal vs. Weight vs. Mass. — As the process of " weigh- 
 ing" makes use of the force called weight, and consists in 
 offsetting the weight of the given substance against that of 
 standard bodies, the resulting quantities of substance are 
 often called " weights " of the substance rather than masses. 
 This usage is sometimes insisted upon as the only one con- 
 sistent with the nature of the process as just sketched. But 
 except in the rarer cases where the object is to ascertain the 
 amount of the forces involved in its action, this usage 
 seems misleading and mistaken. It is true that weight is 
 employed in the operation, but only as means to an end, just 
 as is the beam of the balance or the elastic force called 
 forth in the beam. It is true that the portions of substance 
 measured out have equal weights at the same locality, but 
 we are in general in no wise concerned with that fact except 
 as a convenient incident. What we are seeking is to ascer- 
 tain the quantity of substance with which we are dealing, or 
 to reproduce a stated quantity. This quantity would be 
 the same in any part of the universe, while its weight 
 would vary with its locality even to the extent of vanishing 
 altogether. Even on the earth's surface, the weight of a 
 standard body differs to the amount of one-half of one per 
 cent, an amount quite inadmissible as an error in ordinary 
 scientific work. But if the chemist, for instance, were really 
 to measure off equal weights of a substance (as would be 
 done if the spring balance were used), he would obtain 
 quantities of substance differing to some such extent as this 
 according to locality. In work involving only percentage 
 composition, all weighings being made at the same place, no 
 resultant error would be thus introduced ; but in almost all 
 other chemical and physical measurements it is essential 
 that the results of " weighings " should be comparable at all 
 places and times. Hence it is not with weight that we are 
 concerned. The customary instrument, the equal-arm bal- 
 ance, employed in the usual manner, yields directly what 
 
Chap. Xl] MATTER. 153 
 
 we want, and its results are comparable wherever the obser- 
 vations are made. The quantities resulting from its use are 
 now commonly called '-masses" by physicists. The terms 
 atomic masses, combining viasses, etc., are likewise used in- 
 stead of "atomic weights," etc. 
 
 But notwithstanding the prevalence of the use of the term 
 mass among physicists in recent years, conservative opinion, 
 notably among eminent chemists, is opposed to the employ- 
 ment of that term in place of the time-honored " weight " to 
 denote quantity of substance or matter as measured by the 
 process of weighing. Thus, not only is "weight" still uni- 
 versal colloquially, but it is common with chemists, and is 
 even used by physicists when referring to quantity of sub- 
 stance without special reference to the property of kinergety, 
 with which mass is often confounded ; and further, " weight " 
 is still employed by preference by many eminent authorities, 
 as just indicated ; the terms iveight, atomic weight, combining 
 tveight, etc., being insisted on instead of muss, atomic mass, 
 etc. As the consistent and judicious use of terms is of no 
 less scientific than pedagogic importance, we will review the 
 considerations which may influence the choice in the present 
 instance. 
 
 The trend of the foregoing paragraphs on quantity of 
 matter and the properties of matter shows clearly the sup- 
 posititious character of the assertions that mass is propor- 
 tional directly to either weight or kinergety. To employ 
 mass, therefore, to denote quantity of substance as measured 
 through weighing, is to introduce into terminology an in- 
 ference which can be demonstrated by neither logic nor 
 experiment. This appears to be the sufficient ground for 
 the objection above cited. We may thus justly protest 
 against the assertion of page 147, that " it is only with the 
 inert component [of substance] that we are generally con- 
 cerned when measurements of quantity are to be made, that 
 is with matter." 
 
154 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 The employment of mass wlien referring to quantity of 
 substance with regard to its kinergety is equally objection- 
 able. 
 
 On the other hand, to designate quantity of substance as 
 a "weight" when referring to either the quantity merely or 
 to its kinergety, is equally open to criticism. For the term 
 iveight denotes a force, not a quantity of substance or of 
 matter, and it should be restricted to that denotation. It is 
 the plain duty of science to prevent the perpetuation of the 
 embarrassment now so great to students and members of the 
 technical professions through the prevalent use of " weight." 
 But it is equally clear that mass is not a reasonably accept- 
 able substitute. With full willingness to submit them to 
 the censorship of scientific criticism, the terms weightal and 
 kinergety have here been proposed, — the former to replace 
 the terms mass and zceight in denoting quantity of substance 
 as measured by the equal-arm balance by weighing against 
 the usual standard " set of weights " ; the latter to be em- 
 ployed instead of mass when quantity of substance is re- 
 ferred to with respect to capacity for kinetic energy. The 
 service which the term kinergety can render has, it is hoped, 
 been in some degree made evident through its employment 
 in the pages of this book. As to iveigJital, some further 
 remarks may be added. 
 
 Operations and laws in all branches of the natural sciences 
 are based upon the measurement of quantity of substance by 
 the equal-arm balance or by processes equivalent to this in 
 result. To designate quantities thus measured may it not 
 be a safe and practicable middle course to use the term 
 tveightal ? ^ By derivation, the word is, of course, properly 
 an adjective, a weightal quantity being one taken with re- 
 spect to weight. But. as in the case of " residual " and other 
 like terms, we may properly use the adjective as a substan- 
 tive. Eestricting by definition then the term, as we may, to 
 
 ' A term to which the purist may object. 
 
Chap. XI.] MATTER. 155 
 
 mean only quantities taken with respect to weight measured 
 by the equal-arm balance : — 
 
 Wbightal is Quantity of Substance Measured by 
 THE Equal-Aem Balance. 
 
 The term thus will denote precisely what the physicist 
 and chemist wish to indicate when they refer to quantities 
 of substance now commonly denoted sometimes as " weights," 
 sometimes as "masses." It will, however, denote neither 
 more nor less than is desired, and will introduce no assump- 
 tion, while at the same time serving to remove confusion. 
 The verbal transition from weight to weiglital is so slight 
 as to render the change an eminently practicable one; 
 while the distinction is abundantly sufficient and obvious, 
 whether in the printed or spoken word. Perhaps atomic 
 weightal may find acceptance where atomic weight and atomic 
 mass receive just opposition. 
 
 Molecules and Atoms. — In following out any hypothesis 
 as to the ultimate constitution of bodies, it is found neces- 
 sary to treat very small parts of the body or substance 
 as individual particles. All hypotheses possessing much 
 weight at the present time regard these small portions 
 or particles as distinct aggregations of matter more or less 
 permanent in character ; and the molecular hypothesis as- 
 sumes them in general to be separated from each other by 
 distances usually many times the diameter of the particles. 
 In this molecular hypothesis, these particles are called mole- 
 cules; and the single molecule is regarded as the smallest 
 portion of a substance, whether chemically simple or com- 
 pound, which can exist without loss of the characteristic 
 properties of the substance. The molecule, in its turn, is 
 in general made up of two or more atoms, these being of 
 the same or different kinds, according as the substance is 
 simple or compound. The atom, in its turn, is the smallest 
 portion of an elementary substance which can exist. It is 
 
156 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 not held that the atom possesses, in the same degree as the 
 molecule, all of the properties of the substance. 
 
 It is not uncommon, indeed it is rather customary, to 
 speak of these molecules as molecules of matter. This is ad- 
 missible in the sense that they are composed of matter; but 
 as viewed from the attributes ascribed to them, namely, the 
 possession of various properties, the molecules may possess, 
 or have associated with them, energy of various forms. For 
 instance, the possession of the property of elasticity is 
 ascribed to all molecules and atoms. Until we are able to 
 account for this elasticity in sonae way which proves it to be 
 merely a combination of motion with matter, we shall still 
 be dealing with matter associated with something of whose 
 nature we are utterly ignorant. The vortex-atom theory 
 affords the nearest approach yet made to such an ex- 
 planation. 
 
 Conservation of Matter. — The following generalization is 
 derived from a vast range of experience : — 
 
 The kiiiergety of evenj portion of substance is constant and 
 pennaiicnt. 
 
 If we assume, as is customary, that the kinergety is pro- 
 portional to quantity of matter, that is, to mass, then this 
 proposition may be stated in the form : — ■ 
 
 Tlie mass of every portion of substance is constant and per- 
 manent, i.e. is conserced. 
 
 These statements may be otherwise phrased, as, for in- 
 stance, by saying that the total mass enclosed within any 
 fixed boundary through which no substance can enter or 
 leave during the time under consideration is constant ; or 
 again, substance or matter can neither be created nor de- 
 stroyed. This is one of the most important and far-reaching 
 generalizations of physics. The sort of evidence upon which 
 it rests may be in part indicated as follows : — 
 
 Weightal is proportional to kinergety. Kinergety is con- 
 stant. If a body be separated by any means, physical or 
 
Chap. XI.] MATTER. 157 
 
 chemical, into any number of portions, the total weightal 
 of those portions will be the same as the weightal of the 
 whole before the separation. Therefore the total kiner- 
 gety, and hence the total mass, of all the parts is the 
 same as that of the whole body before the separation. If 
 two or more substances are made to combine chemically, 
 the weightal, and therefore the kinergety and the mass, 
 of the resulting substance or substances is the sum of the 
 weightals, kinergeties, or masses of all entering into the 
 action. Whatever change of properties a body or bodies 
 may undergo, the total weightal of the involved substances 
 at the end of the operation is equal to the total weightal 
 at the outset. This line of proof extends only to such sub- 
 stances as we are able to weigh. The phenomena which 
 involve transference of kinetic energy, or any in which va- 
 riation of kinetic energy of a given body would cause per- 
 ceptible change in the phenomena, are found to be consistent 
 with the general proposition of conservation. The constancy 
 of the periodic times and of the orbits of the planets affords 
 further evidence of the constancy of kinergety and mass in 
 those bodies and in the sun. 
 
 The principle of conservation is generally held to apply 
 not only to matter, but to each element or elementary sub- 
 stance in particular. While we cannot claim this view as 
 demonstrated, still it is generally held, or at least the view 
 that the elements retain their individuality in some degree. 
 If we adopt this point of view, the doctrine of conservation 
 extends to the elements. If we reject it, we must restrict 
 the statement of the doctrine correspondingly, that is, so 
 that it shall apply merely to matter. 
 
CHAPTER XII. 
 COMMENTS ON CERTAIN DEFINITIONS. 
 
 Remarks. — One who suggests departures from estab- 
 lislied custom in scientific definitions, must furnish reason- 
 able demonstration of the validity and the necessity for the 
 proposed changes. Complete proof of the validity of the 
 definitions offered in the foregoing pages is possible only by 
 applying them to all related physical knowledge, ■ — a task 
 impracticable within the compass of a brief and plain expo- 
 sition such as is here undertaken. To the outline of proof 
 afforded by the preceding chapters, it may here be added 
 that such further criticism as the author has been able to 
 bring to bear has not detected any want of sufficiency in 
 the definitions. The necessity for change will be clear to 
 any one who will observe the incoherent and inconsequen- 
 tial character of the definitions current in even the best 
 text-books. Scrutiny of about forty of these has brought 
 conviction on that point. Whatever of clearness and logi- 
 cal sequence the suggested changes have rendered possible 
 in the presentation of the subject-matter of this book, is the 
 best kind of evidence of the need and correctness of those 
 changes. 
 
 It may not be amiss to cite here as a cardinal principle of 
 pedagogics, as well as of science, that no term should be 
 used in science without a clearly assigned denotation, that 
 is, without having been clearly defined. However broad 
 and vague is the connotation of the term, its denotation 
 (p. 7) must be perfectly sharp. This is always possible, al- 
 
 158 
 
Chap. XII.] COMMENTS ON CEUTAIN DEFINITIONS. 159 
 
 though often neglected and sometimes denied, — notably 
 in the definition of "matter." It is true that we know- 
 little or nothing of the nature of matter, but we are not 
 therefore unable to give to the term a perfectly precise 
 definition. If we were, the term should have no place in 
 science. We employ it to stand in place of a definite idea 
 which we can state (cf. p. 10), and that statement con- 
 stitutes the desired definition. The sundry attributes form 
 no part of the definition or denotation. The more the 
 nature of matter becomes known, the better can its defini- 
 tion be made, of course, but this is no excuse for present 
 lack of clearness. Accuracy in the definition and employ- 
 ment of terms is a prime duty imposed by considerations of 
 economy of effort. 
 
 On Definitions of Matter. — To define matter as " that out 
 of which bodies are made up " fails in two chief respects. 
 First, this denotation includes energy as well as matter; 
 for every known body contains as essential components 
 many forms of energy. No theory of matter even exists 
 which does not ascribe to the ultimate particles of substance 
 some one or more forms of energy, or forces which must be 
 due to energy. In some of the theories, the force or energy 
 is boldly engrafted in the desired form upon the inert mat- 
 ter, in others, a special mode of motion is imposed upon the 
 matter by which it is aggregated into separate, discrete 
 particles of substance. In all cases, the energy is there. 
 Second, this class of definitions fails to indicate the infer- 
 ential character of the idea denoted by matter. 
 
 Perhaps the most strongly intrenched definition is that 
 given by the phrase " that which occupies space." Specula- 
 tions on the nature of matter strongly support the view that 
 this may ultimately prove to be the correct or at least a 
 partial definition. But for the present, one may perhaps 
 be permitted to question its availability. Bodies occupy 
 space. But with a degree of certainty at least equal to our 
 
160 FUNDAMENTAL PHYSICAL CONCEPTS. [Pakt I. 
 
 knowledge of their existence, we know that this extension 
 is due in large part to the energy which kee^DS their par- 
 ticles at a distance from each other. There is indeed no 
 direct evidence that matter devoid of energy does occupy 
 space. The view that it does so is purely a hypothetical de- 
 duction. This fact alone is enough to debar this definition 
 from use in elementary physics. A definition based upon 
 a remote, specific hypothesis not presentable to a beginner 
 is certainly not adapted for use in elementary physics. 
 
 Some modern text-books follow Maxwell's lead of thirty 
 years ago, and define, or rather describe, matter as the 
 " vehicle of energy." With a qualification to indicate the 
 inferential character of the idea of energy, this may serve 
 fairly well, although lacking something in definiteness. 
 But it presupposes an adequate defi.nition of energy, which 
 is rarely if ever provided. 
 
 On Definitions of Energy. — The current definition of energy 
 is usually given in the words, — Energy is the capacity of 
 doing work. Therefore in any orderly presentation of the 
 subject, a definition of work must precede that of energy. 
 But the accepted definitions of work are either of the form, 
 — Work is the action of moving a body against or by a 
 force ; or of the form, — -Work is any process of transference 
 or transformation of energy. These have been fully and 
 formally given in the chapter on Work. Now force can 
 be adequately defined only in terms of energy, as at page 
 41, and as will be further shown in the next section. Thus 
 the definition of energy in terms of work leads merely to 
 the vicious circle, — energy, work, energy. It must not, 
 however, be supposed that any such deficiency in logic is 
 openly spread upon the pages of the text-books. The defi- 
 nitions and explanations of force there given furnish the 
 student with so inadequate a view of its nature that he is 
 generally quite unaware of its real relation to energy, and 
 thus fails to recognize the fallacy. 
 
Chap. XIT.] COMMENTS ON CERTAIN DEFINITIONS. 161 
 
 The claim might be urged that it would be possible to 
 define work in such terms as to remove this difficulty ; and 
 that to do so would be desirable, since it would reenforce 
 the above long-established definition of energy. However 
 advantageous this might be, it appears not to have been 
 undertaken, and its success is at least doubtful. For in- 
 stance, work might be said to be the operation of changing 
 the state of motion of a body or of maintaining its motion 
 when not free. This covers all cases of work, and in so far 
 would suffice. But some difficulty might be found in suit- 
 ably defining " free," and apart from this the suggested 
 form is easily seen to be far inferior to the existing ones in 
 adaptation to quantitative measurement. 
 
 The formulation of energy as the power to change the 
 state of motion of a body, is free from the logical and other 
 embarrassments attaching to the current definition, and to 
 the suggested change in the definition of work. It expresses 
 the concept of energy in terms so simple as to be intelli- 
 gible even to persons unfamiliar with physics, ■ — a point of 
 some importance in the extension of the correct colloquial 
 use of the word. Not less important is this simplicity in 
 the scientific presentation of the subject, as it permits the 
 introduction of the idea of energy in advance of that of 
 force, or work, or even of matter. That this early introduc- 
 tion is not only logical but of great help in the elementary 
 presentation of physical science, seems not to be duly recog- 
 nized. It is hoped that the latter point may have received 
 some illustration in the first part of this book. That the 
 leading position in sequence as well as in importance logi- 
 cally belongs to energy, maybe briefly indicated as follows. 
 Everything which we observe, that is, all phenomena and 
 observed properties, are due to energy or its changes of 
 form or location. All experience has therefore to do with 
 energy, and with that alone. As it is foremost and ubiqui- 
 tous in experience, so should it also be, if possible, in the 
 
162 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 treatment of physics. Two or three decades ago, this was 
 impossible ; but it is so no longer. Porce and work must 
 now be dealt with as actions of energy, and matter must be 
 regarded, for the present at least, as an inference one step 
 more remote than energy. 
 
 The fact that the majority of phenomena are not recogniz- 
 able as due to " change in state of motion of bodies," might 
 suggest itself as indicative of inadequacy in that form of 
 definition. But although we do not know that such phe- 
 nomena are due to change of motion, we are equally unable 
 to show that they are not so. Moreover, it is not at all es- 
 sential that the denotation of a term should cover all the 
 attributes of the term, that being the function of the conno- 
 tation. The criticism would therefore not be justified. The 
 adequacy of the definition can be called in question only by 
 its failure to include as energy anything otherwise correctly 
 demonstrated to be energy ; by failure to exclude anything 
 known otherwise not to be energy ; by leading to inferences 
 contradictory to known facts ; or by failure to lead up to a 
 rational primary method of measuring quantity of energy. 
 Sufficient evidence on these points has been given. The 
 necessity noted in the last of them appears to be the cause 
 of a tendency in some careful writers to advocate defini- 
 tions which are really denotative not of energy but of quan- 
 tity of energy. A similar tendency is much more pronounced 
 in the case of force. Such a practice, however harmless in 
 mathematical physics, is inadmissible in a fundamental 
 treatment. 
 
 To define energy as that which is (or through its changes 
 is) the cause of all physical phenomena, is unsatisfactory. 
 The idea conveyed is too vague and intangible to be of service 
 to the vininitiated, and fails to lead directly to any primary 
 method of energy measurement. Its definition as, — That 
 which underlies physical phenomena and which is constant 
 in quantity throughout its many changes in form, is simi- 
 
Chap. XII.] COMMENTS ON CERTAIN DEFINITIONS. 163 
 
 larly unsuitable. Both of these are descriptions rather than 
 definitions. 
 
 On Definitions of Force. — Two centuries ago, Newton em- 
 ployed the term vis impressa with entire clearness to de- 
 note precisely the kind of quantity now generally termed 
 force. Newton's definition of the term is here quoted from 
 the " Principia " : — 
 
 Def. IV. — " Vis impressa est actio in corpus exercita, ad 
 mutandum ejus statum vel quieseendi vel movendi unif ormi- 
 ter in directum." 
 
 Somewhat literally rendered this is : — 
 
 Def. IV. — Impressed force is action exerted upon a body 
 to change its state either of resting or of moving uniformly 
 in a straight line. 
 
 The term vis impressa and its translation force and im- 
 pressed force are now given up by universal consent to the 
 denotation rendered classic by the " Principia," but until a 
 few decades ago the three terms, vis, force, and Kraft, now 
 properly held to be equivalent, were loaded with the added 
 significance now denoted by energy, — at least so far as the 
 latter concept was then apprehended (cf. p. 76). In the 
 somewhat sentimental revulsion of feeling which has at- 
 tended the effort to establish the proper discrimination 
 between the denotation of force and energy, there has been 
 manifested a disposition to deny to force any adequate rec- 
 ognition. Before dealing with this point, however, let us 
 examine Newton's definition in detail. 
 
 The words statum vel quieseendi vel movendi uniformiter 
 in directum are the exact equivalent for the phrase " state 
 of motion " as defined at page 15. Newton's definition may 
 then be rendered by : — 
 
 Force is that action exerted on bodies to change their state 
 of motion. 
 
 In the employment of actio, Newton must have intended 
 a clear distinction between action and agent. Also he ap- 
 
164 FUNDAMENTAL PHYSICAL CONCEPTS. [PaktI. 
 
 pears to have used actio in the sense in which action has 
 been employed in the deiiuition of force in this book (p. 41) ; 
 namely, as denoting an operation. It is therefore pertinent 
 to inquire, — " action " of what ? It seems altogether im- 
 probable that Newton possessed an adeqviate answer to this 
 question, or that such an answer could have been given by 
 any one before the middle of the present century. About 
 all that modern science has contributed to the extension of 
 the concept of force lies in that answer, and yet this contri- 
 bution, great and necessary as it is, is but indistinctly sug- 
 gested in even recent text-books of physics. Action of 
 what ? Action of energy. Insert then the words of en- 
 ergy in the Newtonian definition, and it then covers the 
 modern concept of force. But it also covers more than that 
 concept; it includes also the concept of work. Newton's 
 phraseology "to change its state," etc., includes both the 
 production of a tendency and of the change also. Now the 
 latter action is work. Hence the necessity for the introduc- 
 tion of the word tendency, in the proposed definition. If 
 the foregoing comment is substantially correct, the employ- 
 ment of the Newtonian definition at the present time with- 
 out modification is to ignore the entire progress of two cen- 
 turies in this direction. 
 
 A distortion of Newton's definition, to which attention 
 should be directed, consists in the substitution of "that 
 which " for action. Thus modified, the words form a defi- 
 nition of energy, not of force, and have been far from help- 
 ful to the student. 
 
 Force has also been defined as "tendency to acceleration." 
 But this formula, like the others already referred to, fails 
 to indicate the known relation of force to energy. Also it 
 confounds the action, which is force, with the result, which 
 is not force but is the effect produced by the acting energy 
 through its action, force. The employm'ent of " accelerar 
 tion " instead of " change in state of motion " is admissible, 
 
Chap. XII.] COMMENTS ON CERTAIN DEFINITIONS. 165 
 
 on the ground that all change in state of motion appears as 
 acceleration when referred to a proper point (cf. p. 15). 
 But the brevity gained by the substitution is more than 
 offset by the loss of directness in application to phenomena. 
 For many years, two other force definitions which are 
 equivalent to one another have been current, although by no 
 means universally acknowledged to be adequate. They state 
 force to be, — "the space-rate of transference of energy" 
 and " the time-rate of change of momentum of a body." The 
 fundamental defect in both is that they define, not force, but 
 quantity of force. The practice of employing definitions of 
 quantity in place of those of the abstract concept has found 
 many followers, but one may be permitted to doubt whether 
 the principle involved has been duly recognized, or indeed 
 even the bare fact itself. The definition of anything 
 which is capable in the concrete of quantitative measure- 
 ment should, to be sure, lead up directly to a consistent 
 primary method for that measurement; but this proposition 
 in no way justifies the principle of substitution above stated, 
 nor does there seem to be any justification therefor. More- 
 over, the space-rate and time-rate definitions of force fail in 
 two other ways. First, they assign to force no recognition 
 as a physical action, when motion is not in progress. In 
 other words, they ignore the existence or occurrence of 
 force in static phenomena. Of course the text-books which 
 thus deny recognition to balanced forces in the definition 
 do not consistently omit the discussion of them in the phe- 
 nomena in which they occur. Second, these definitions are 
 further lacking in that they ignore, and by implication 
 deny, that there is any interpretation to be given to force 
 other than a purely mathematical one. Such a position is 
 not consonant with the generally adopted attitude of mind 
 towards physical phenomena, energy, and matter. The 
 concepts, respectively, of a thing, of that thing in action, 
 of an action by that thing, and of an effect produced by 
 
166 FUNDAMENTAL PHYSICAL CONCEPTS. [Part I. 
 
 the thing through the action are perfectly familiar and 
 rational. The relation between these three is precisely the 
 relation between energy, force, and the effect of the energy 
 through this action, whether the effect be change of motion 
 or merely tendency to that change. It is just this relation 
 which the mode of presentation adopted in this book will, 
 it is hoped, render more clear ; as from this standpoint the 
 concept of force becomes duly subordinated to that of en- 
 ergy, and falls naturally into its useful subsidiary place. 
 
Part II. 
 
 SPECULATIONS AND THEORIES ON MATTER 
 AND ENERGY. 
 
" Natural Science considers the world a mechanism, and for that 
 purpose transforms the reality in a most complicated and ingenious 
 way. It puts in the place of the perceivable objects unperceivable 
 atoms which are merely products of mathematical construction quite 
 unlike every known thing ; and nevertheless these atoms are scien- 
 tifically true, as their construction is necessary for that special logical 
 purpose. To aflirm that they are true, means that they are of objec- 
 tive value for thought. But it is absurd to think, with the material- 
 istic philosopher, that these atoms form a reality which is more real 
 than the known things, or even the only reality, excluding the right 
 of all not space-filling realities. . . . There is, indeed, no physical 
 object in the world which natural science ought not to transmute into 
 atoms ; but no atom in the world has reality ; and these two state- 
 ments do not contradict each other. 
 
 " . . . [It is not wiser than] to cast up against the physicist that his 
 moving atoms do not represent the physical world because they have 
 no color and sound and smell. If they sounded and smelled still, the 
 physicist would not have fulfilled his purpose."- — Hugo MtiNsxER- 
 BERG, in The Atlantic Monthly, May, 1898. 
 
CHAPTER I. 
 FUNCTION OF THEORY AND HYPOTHESIS. 
 
 Value of Theory. — In view of the diversity of the ele- 
 ments and of the forms of energy, it is entirely within 
 the province of physics to inquire in what this diversity 
 consists, and what is the nature of matter and of energy, 
 the answer to the latter involving the explanation of force. 
 
 Hypotheses, even imperfect and insufficient, are not only 
 helpful, but in fact are the only stepping-stones by which 
 we can advance in this direction. Their function is to 
 point out a possible relation between a multitude of ob- 
 served phenomena, which shall serve to unite these into 
 a connected system. Such a system, although it may be 
 wide of the truth and may prove ultimately to be entirely 
 incorrect, may yet be of the utmost service for the time 
 being. The more general view of the phenomena thus ob- 
 tained enables the mind better to grasp them in their rela- 
 tion to each other, and suggests lines of investigation which, 
 if carried out, contribute just so much to scientific knowl- 
 edge. These new studies will accumulate added facts 
 which will either confirm and extend or contradict the hy- 
 potheses which led to their discovery. In either case, 
 progress has been made, and the usefulness of a hypothesis 
 may be as well demonstrated by its overthrow as by its 
 confirmation. Value of theory to the student of physics, 
 through its cultivation of the scientific use of the imagina- 
 tion, is of the highest importance ; and the continual devel- 
 opment of novel suggestions and possibilities reduces the 
 danger of narrow or biased interpretation of fact. The 
 
 169 
 
170 SPECULATIONS AND THEORIES. [Part II. 
 
 remarkable stimulus to the development of new branches 
 of mathematical inquiry, through their application to phys- 
 ical theories, bears further testimony to the usefulness of 
 such theories. As an example of these three points, one 
 has but to consider the vortex-atom theory, and to contrast 
 one's own thoughts upon the nature of matter before and 
 after a careful study of this theory. But in order that 
 hypotheses, and theories which are developed from hypoth- 
 eses, should possess their highest efficiency, it is requisite 
 that they should be regarded from an entirely judicial 
 point of view. No theory can ever deserve that order of 
 credence which attaches to an observed fact. All theory 
 is based on logical inference, in the process of which are 
 always involved some propositions or hypotheses which are 
 incapable of absolute demonstration. All theories must, 
 therefore, be regarded as at best tentative ; and when the 
 narrow range of human knowledge is recognized, they cer- 
 tainly must be held to be entitled to a very small degree 
 of real confidence ; they certainly can be no more than the 
 crudest approximation to the ultimate truth. If this esti- 
 mate be placed upon their value and they be employed 
 merely as temporary structures to be replaced by better at 
 the earliest possible moment, they may serve a most help- 
 ful purpose. This is certainly the only scientific view to 
 be taken of them, and those persons who found upon the 
 existing knowledge of the physical universe, — a knowledge 
 in which theory plays so large a part, — a superstructure 
 of materialism, must certainly be considered as utterly un- 
 scientific. To deduce materialism from physical science 
 demands either an illogical mind or procedure, or the addi- 
 tion to that science of some propositions not pertaining to it. 
 Purpose of this Section. — The following summaries, hy- 
 potheses, and theories are grouped here for two purposes : 
 first, to show the general trend of speculation by the most 
 eminent physicists, upon the nature of matter and of en- 
 
Chap. I.] FUNCTION OF THEORY. 171 
 
 ergy ; second, to indicate thus certain ways in which energy 
 may be supposed to produce force or, in other words, to de- 
 scribe briefly certain mechanical operations which have 
 been imagined as possibly representing that action of en- 
 ergy which we call force. This second point is really em- 
 braced in the first. The manner in which the expansive 
 force of gases is explained, under the kinetic theory of 
 gases, affords one sucli illustration. This is shown in an- 
 other phase in the vortex-atom hypothesis. The latter 
 theory gives a radically new view of the possible nature of 
 chemical action. As a whole it also affords an unprece- 
 dented example of the deduction of most diverse laws and 
 phenomena by pure mathematics, starting with a supposed 
 material system of the utmost simplicity of character. No 
 student of physics can afford to deny to these various hy- 
 potheses a most careful consideration. It is by far too 
 customary to regard them slightingly because of the recog- 
 nized improbability of their ultimate truth. 
 
CHAPTER II. 
 
 KINETIC THEORY OF GASES. 
 
 Statement of the Theory. — If a perfectly elastic, spheri- 
 cal ball in rectilinear motion strikes against a fixed, per- 
 fectly elastic, plane, frictionless surface, it rebounds from 
 that surface with unchanged velocity and in a direction in 
 accordance with the law of reflection of light. The re- 
 bounding ball thus possesses precisely the same amount of 
 kinetic energy as before impact. It has, therefore, im- 
 parted none of its energy to the surface with which it 
 collided, but during the entire time of contact (this has a 
 sensible duration) a stress existed between the ball and 
 the surface. During the first half of this collision the 
 kinetic energy of the ball is being transformed into elastic 
 energy in the ball and in the surface. This energy is ex- 
 erting a stress between the ball and the surface. During 
 the second half of the collision the stress is maintained by 
 their elastic energy, and the stress at any instant is equal 
 to the space-rate at which energy is being interchanged be- 
 tween the two surfaces. Let us consider for simplicity the 
 case where the ball strikes normally upon the surface. 
 
 If we were to consider the ball and surface merely as 
 elastic substances without questioning the detail of the 
 mechanism of this elasticity, we should describe a collision 
 in this way : — As the ball comes into contact with the sur- 
 face, its kinetic energy is gradually transformed into energy 
 of elasticity within itself and within the surface with which 
 it collides. This elastic energy is called into play by the 
 change in form of the ball, which becomes flattened in the 
 
 172 
 
Chap. II.] KINETIC THEOBY OF GASES. 173 
 
 direction of motion, and that of the surface, which becomes 
 to a certain extent indented. Daring the time in which the 
 ball is being brought to rest an increasing stress is exerted by 
 this elastic energy between the ball and the surface. When 
 the ball has come to rest, all this kinetic energy has been 
 transformed. The stress between the surfaces and between 
 the parts of the ball, as well as between the parts of the 
 surface, is now everywhere at a maximum. The elastic 
 energy producing this stress now sets the ball gradually in 
 motion away from the surface, and the process is of such 
 a nature that by the time all of the particles have returned 
 to their normal positions the ball is again moving with its 
 former linear velocity in all parts, and all parts of the sur- 
 face have again come to rest in their normal position. Con- 
 sidering the force relative to the ball during the collision, 
 it is a pressure by the ball during the first half and upon the 
 ball during the second half of the impact. The bodies being 
 supposed perfectly elastic, no energy remains in other forms. 
 To hold the surface in position against such a collision, 
 it is clear that it must be supported by a force which at 
 every instant is equal to the stress between the surface 
 and the ball ; otherwise the surface will move. This 
 supporting force must evidently be in the same direction 
 throughout the contact. If the collision is in a direction 
 normal to the surface, then the supporting force is also nor- 
 mal to the surface and towards the ball. If the collision 
 is oblique, the supporting force is still in the same direction 
 as before, unless there is friction between the points of con- 
 tact, because the component of the ball's velocity parallel 
 to the surface remains unchanged by the collision, and the 
 force exerted by the ball upon the surface has, therefore, no 
 component in a direction at right angles to the normal. 
 The stress between the ball and the surface varies from 
 instant to instant, so that the supporting pressure will vary 
 in like measure. We thus see how a spherical, elastic ball 
 
174 SPECULATIONS AND THEORIES. [Part H. 
 
 entering into collision with a plane, solid, elastic surface 
 produces a stress in the direction normal to it. If such a 
 surface be supported by the hand, we shall experience a 
 muscular sensation corresponding to the muscular effort 
 which we exert. In all such collisions there will be certain 
 minor changes in distribution of energy which have not 
 been indicated. In the normal collision, a wave of vibra^ 
 tion energy of small intensity will be transmitted in all 
 directions through the restraining surface from the point of 
 contact, so that the rebounding ball will in fact not carry 
 oil its whole initial amount of kinetic energy, and moreover 
 will possess a definite, periodic vibration, which wiU consist 
 of a periodic lengthening and shortening of the diameter in 
 the direction of motion, and an accompanying corresponding 
 change in other diameters. The kinetic energy will be less 
 after than before impact by the sum of the energy of vibra- 
 tion of the ball and the vibration energy left in the surface. 
 If the collisions are oblique, the ball and surface will simi- 
 larly acquire vibration ; and if there is friction between the 
 ball and surface during the time of collision, the ball will 
 also acquire rotation, so that in addition to the energy of 
 vibration there will be a certain amount of energy of rota- 
 tion to be deducted from the kinetic energy of the returning 
 ball. These secondary actions involve usually but a small 
 portion of the energy of a single impact, and their effect 
 will be further considered in dealing with the phenomena 
 produced by the collision of a large number of balls en- 
 closed in an elastic receptacle. 
 
 Imagine now a continuous and uniformly distributed 
 shower of such balls striking normally upon a small sur- 
 face and rebounding without collision with each other. At 
 any instant several or many of these balls will be in colli- 
 sion with the surface. But as any one is equally likely to 
 be in any possible phase between the initial and the final 
 state of the collision, a variety of phases will exist at any 
 
Chap. II.] KINETIC TIIEOBY OF GASES. 175 
 
 given instant. As the density of such a shower is greater 
 (that is, the number of particles impinging on a unit of 
 area), more and more phases will be represented at any- 
 instant in the colliding balls ; and ultimately all phases 
 will be equally represented at every instant. Thus, the 
 total pressure on the surface will be more and more nearly 
 uniform the greater the density of the shower. If instead 
 of the normal shower, the surface be struck by a shower of 
 balls proceeding equally in all directions, oblique as well as 
 normal, the stress will still be a normal one, and will, as 
 before, be more nearly constant the greater the number of 
 balls. If there is no friction between the balls and the 
 surface, there will be no force in the plane of the surface ; 
 but if there be friction, the amount of such force in all di- 
 rections will be equal, and its resultant will therefore be 
 zero. 
 
 Imagine now a box of any size and shape with walls of 
 perfectly elastic material and containing within it nothing 
 (i.e. a perfect vacuum) except an immense number of ex- 
 ceedingly small, spherical, elastic balls such as have just 
 been considered. Let these balls or particles be so minute 
 that their number per cubic inch, if distributed uniformly 
 through the volume of the box, woTild be reckoned by 
 millions or billions, but let their size be so small that 
 their total volume is exceedingly small compared to the 
 capacity of the box ; that is, suppose the capacity of the 
 box many million times the total volume of the particles. 
 
 Imagine the box to be shaken for some time with great 
 violence and then brought to rest, what will be the charac- 
 teristics exhibited by the space thus filled with moving 
 particles ? 
 
 The motion of such an imaginary system has been mathe- 
 matically investigated by Clausius, Maxwell, Lord Kelvin, 
 Joule, Boltzmann and many others. It has been shown 
 that the system of particles, whatever might be the initial 
 
176 SPECULATIONS AND THEORIES. [Part II. 
 
 state of motion produced by the agitation above described, 
 would ultimately and rather promptly settle down into a 
 condition of kinetic equilibrium. This final condition at 
 any point within the volume, and at any given instant of 
 time, is such that a sensibly equal number of particles is 
 moving in all directions. If the box and contents were at 
 the same temperature, no kinetic energy would be given up 
 to the box by the particles, but the total energy of the 
 system of the particles would remain a constant, provided 
 that the box transfers none of its elastic energy to sur- 
 rounding bodies and that no energy of any other form were 
 introduced through the box. The particles in their motion 
 would, on the average, traverse in straight lines a distance 
 very long in proportion to their diameters, but would be 
 continually coming into collision with other particles or 
 with the sides of the box. Prom these encounters with 
 each other, the particles would rebound with changed veloci- 
 ties and directions, and therefore, in general, each with dif- 
 ferent kinetic energy, but the total energy of the colliding 
 particles would be unaltered by the collision. The particles 
 striking the side of the box would rebound, also with un- 
 changed energy on the average, although in any individual 
 collision there might be a greater or less change, owing to 
 vibratory energy in the side of the box at point of contact. 
 Each particle would possess not merely translatory motion 
 but also rotation and vibration. Some of the particles at 
 any given instant would be moving with very high velocity, 
 some with very low velocity, and the larger proportion 
 with intermediate velocities. Considering any one particle, 
 the distance which it traverses between successive encoun- 
 ters would be sometimes long, sometimes very short ; and 
 the direction of its free path in these intervals would be 
 changed at every collision. Its amount of energy of rota- 
 tion and of vibration would likewise be changed at every 
 collision. Although all these changes are continuously 
 
Chap. II.] KINETIC THEOBY OF GASES. 177 
 
 going on in the individual particles, so that it is impossible 
 to trace mathematically the history of the individual, still. 
 Maxwell has shown how the particles might be treated by a 
 statistical method, and has thus discussed the nature of the 
 motion and has shown that the system would be governed 
 by laws corresponding closely, in many cases exactly, to the 
 known laws of gases. Boltzmann has shown by this method 
 that a state of kinetic equilibrium will come about as above 
 stated, in which in spite of the continual individual changes, 
 the proportion between the total kinetic energy (heat), 
 the total energy of vibration, and the total energy of rota- 
 tion, in the system will be a constant. He has shown that if 
 any disturbance of this proportion is introduced, a readjust- 
 ment will spontaneously occur to bring back the former 
 distribution of energy. This state of kinetic equilibrium is 
 therefore a stable one. 
 
 It has been shown mathematically th^t such a system of 
 particles would possess nearly, if not quite all, of the prop- 
 erties exhibited by a perfect gas. By their bombardment 
 upon the sides of the box, the particles would produce a 
 pressure in every way analogous to gaseous pressure. 
 Across any imaginary plane surface, at any point within 
 the gas, there will be a transfer of particles. If this plane 
 were to become solid, the transfer would be changed to a 
 bombardment which would produce upon either side of the 
 plane a pressure of the same intensity as upon the walls of 
 the box, thus illustrating the internal pressure of a gas. If 
 the portions on either side of this imaginary plane were to 
 move over one another, they would experience internal 
 friction corresponding to the internal gaseous friction, or 
 viscosity, described at page 49. Any one particle would 
 move gradually and irregularly from one to another part of 
 the box, and the nature and laws of this motion would cor- 
 respond precisely with those of gaseous diffusion. The 
 mean square of the velocity in such a system would be pro- 
 
178 SPECULATIONS AND THEORIES. [Part II. 
 
 portionate to the temperature of the gas. And the laws of 
 Boyle and Dalton follow directly as properties of the system. 
 The pressure exerted by the system per unit of area of sur- 
 face exposed would be normal to the surface and propor- 
 tional to the average kinetic energy of the particles ; and 
 two such systems, each composed of particles of the same 
 size and mass as those of the other system, or of different 
 sizes or masses, would be in thermal equilibrium when 
 the average kinetic energies of the particles of the two 
 systems were equal. If two boxes containing each a dis- 
 tinct system were to be brought end to end, and the parti- 
 tions removed so that the systems would be in direct contact, 
 then, if the pressures were equal, there would be no result- 
 ant transfer of kinetic energy (that is, of heat) across the 
 junction in either direction, provided also that the average 
 kinetic energy of the particles of the two systems were the 
 same. If the two systems were mixed together, the same 
 relation would hold when the normal distribution of the 
 particles was attained. 
 
 Maxwell has shown that this condition of equality of 
 kinetic energy is the condition for temperature equilibrium, 
 not merely between such systems as those above described 
 which correspond to different gases, but also between any 
 two molecular systems, as between solids and liquids, or 
 either of these in contact with gases. This point seems to 
 be one of fundamental importance, since without it there 
 would remain to be shown a reason why the kinetic energy, 
 that is, the heat of an enclosed gas, should not be carried 
 away in the form of waves of vibration energy through the 
 enclosing surfaces however perfect their elasticity. This 
 point appears not to have been generally considered in deal- 
 ing with the theory of gases. 
 
 The kinetic theory of gases is based on the assumption 
 that the gas is constituted of perfectly elastic particles, 
 whose dimensions are excessively small compared with the 
 
Chap. II.] KINETIC THEORY OF GASES. 179 
 
 volume of the gas, and "whose free paths are very long com- 
 pared with the diameter of the particles. In its simplest 
 form it makes no further assumption regarding these par- 
 ticles than those which are contained or implied in the 
 above-described system of moving spherical particles, and 
 the nature of the particles of the gas is supposed to be 
 similar in all respects to that of the system of elastic par- 
 ticles just described. Most of the properties of perfect 
 gases can be accounted for by this hypothesis, and with a 
 few additional hypotheses or considerations, the deviations 
 of gases from the simple laws, as they approach their points 
 of condensation, can be more or less fully accounted for. 
 Some phenomena seem to call for additional hypotheses, 
 ascribing to the particles forces of mutual attraction or 
 repulsion, or both, varying according to arbitrarily assumed 
 powers of the distance between the centers of the particles. 
 Of course such an addition complicates the theory in a high 
 degree, and these assumed forces in turn demand an explar 
 nation ; still, even in this form, the kinetic theory has 
 proved of the utmost value in advancing our knowledge, 
 both theoretical and experimental, of the laws of gases. 
 Many of the deviations, however, are accounted for much 
 more simply and with a high degree of approximation by 
 considering the influence upon the laws of the supposed 
 system which would arise from assigning to the particle a 
 greater size in proportion to their average distance apart. 
 
 From various data it has been found possible to calculate 
 approximately the average velocity of the molecules of any 
 gas at ordinary temperature, the mean length of free path, 
 and very roughly the apparent diameter of the particles or 
 molecules. In the article " Molecule " in the Encyclopedia 
 Britannica, Maxwell has tabulated values for several gases. 
 As belonging to a familiar gas and one which is a large 
 constituent of air, the following values for oxygen are 
 quoted. 
 
180 SPECULATIONS AND THEORIES. [Part II. 
 
 Mass of Molecule (hydrogen = 1) = 16. 
 
 Velocity (of mean square) at 32° F. = 465 meters per 
 second (= 18000. inches or 1500. feet per second). 
 
 Length of mean free path = . 00 000 0056 meters (= 0.00 
 000 22 inches). 
 
 Collisions of an average particle per second =7 600 000 000. 
 
 Diameter = 0.00 000 000 076 meters ( = 0.00 000 0030 
 inches). 
 
 Mass of Molecules = 0.00 000 000 000 000 000 000 0074 
 grammes (= 0.00 000 000 000 000 000 000 0005 grains). 
 
 Lord Kelvin has discussed the conditions of equilibrium, 
 and the various properties, of systems of such particles 
 grouped in more or less complicated ways. He has inves- 
 tigated the condition of stability of such groups, and of 
 mixtures of groups of different complexities, the individual 
 particles of the group being held together by mutual forces. 
 He has thus been able to arrive at systems which would 
 exhibit many of the peculiar properties, both mechanical 
 and electrical, of liquids and even of solids. He has also 
 been able to find analogies for the case of a liquid or solid 
 in contact with its own vapor. 
 
 Many other attempts have been made to extend these 
 kinetic explanations of the structure of solids and liquids, 
 but the difficulties are here much greater than for gases, 
 owing to the close proximity existing between the particles, 
 and to the play of the mutual forces, which is negligible 
 in gases because of the great distance between the par- 
 ticles. 
 
 An excellent presentation of the kinetic theory of gases 
 may be found in Maxwell's "Theory of Heat," Watson's 
 " Kinetic Theory of Gases," and Risteen's " Kinetic Theory 
 of Gases." No one who wishes to follow out the literature 
 of the subject should fail to read the various contributions 
 to it by Maxwell (" Scientific Papers " and his articles in 
 the Encyclopedia Britannica), by Lord Kelvin ("Mathe- 
 
Chap. II.] KINETIC THEORY OF GASES. 181 
 
 matical and Physical Papers "), and by Clausius (articles 
 in Poggendorff's "Aunalen "). 
 
 Comment. — Of the kinetic theory of gases, and of the cor- 
 responding molecular theories of liquids and solids, it may 
 be said that they have served and continue to serve a most 
 useful purpose in the true function of theory. They are, 
 however, not ultimate in character. The assumed perfectly 
 elastic molecule, with its attendant supposed forces and 
 properties, plays satisfactorily the role of a unit in a great 
 company whose effect as a whole only is to be considered. 
 To some extent it is effective when treated as an individual 
 also. But as far as any ultimate explanation or even sim- 
 plification of the forces, forms of energy, or views of matter 
 are concerned, it is of small effect. The molecule, as its 
 name implies, is merely a little body. The molecular theo- 
 ries offer no account of energy, of elasticity, or of the energy 
 which must be present in the molecule or atom to give it 
 shape, hardness, etc. It is left for the vortex-atom theory 
 to take the first step toward the ultimate analysis of the 
 properties of substance. 
 
CHAPTER III. 
 
 LE SAGE'S THEORY OF GRAVITATION. 
 
 Statement. — In the year 1782 Le Sage of Geneva pub- 
 lished a hypothesis or theory, to explain the action of 
 gravitation. Somewhat similar, but much less complete 
 and careful explanations had been previously suggested, 
 and to them Le Sage makes conscientious reference. The 
 theory has been modified by others, but a sketch of it in 
 the original form, omitting for brevity some of the limitar 
 tions which Le Sage so carefully stated, will be here given. 
 
 Imagine an indefinitely large space containing nothing 
 except streams of minute particles (or as Le Sage calls 
 them " corpuscules ") flowing through the space in a multi- 
 tude of directions. Each stream consists of particles flow- 
 ing parallel to each other, uniformly distributed, having an 
 enormous velocity. Each stream is supposed to have 
 a sectional area sufficient to cover all parts of the space 
 under consideration. The directions of the various streams 
 are supposed to be uniformly distributed, so that all direc- 
 tions are equally well represented. The particles are sup- 
 posed to be 2xrfectly hard and inelastic, and of a size so 
 small relatively to the space between them that collisions 
 between the particles of the various intersecting streams 
 are exceedingly rare — a condition which is aided by the 
 enormous velocity ascribed to the particles. 
 
 At any point within this space, then, imagine a small 
 sphere to be described. This sphere would be bombarded 
 equally on all sides by the corpuscules. If the sphere 
 were permeable by the particles, they would simply pass 
 
 182 
 
Chap. III.] LE SAGE' S THEORY OF GRAVITATION. 183 
 
 through, it. If it were wholly impermeable, they would 
 either be brought to rest by it, if the sphere were inelastic, 
 or they would rebound from it if the sphere were perfectly 
 elastic. In either case, the sphere would experience an 
 equal pressure from all sides, and would therefore be in 
 equilibrium in the midst of the moving particles. 
 
 Suppose now a second sphere near the first. If both were 
 entirely permeable they would be in equilibrium, and no 
 motion of either would ensue. If, however, they were 
 wholly impermeable, then each would arrest the particles 
 coming against it from the side turned away from the other 
 sphere, and each would thus cast a sort of shadow upon the 
 other. Thus sphere number one would receive less bom- 
 bardment on the side next to number two, and similarly 
 number two would have less bombardment on the side 
 towards number one. The pressure, therefore, upon the 
 sides facing each other would be less than the pressure on 
 the outward-facing side, and each sphere would, therefore, 
 experience a force whose direction it is easy to see would 
 be towards the other sphere ; in other words, the spheres 
 would " tend to approach " one another. 
 
 Le Sage assumed the corpuscules to be perfectly inelastic. 
 Suppose also, for the moment, that the spheres are perfectly 
 impermeable. It will be clear, on a little reflection, that 
 the amount of the force upon each sphere will be, in this 
 case, proportional to the product of the equatorial sections 
 of the two spheres. Por the moment, consider each sphere 
 to be replaced by an impermeable, circular, plane surface 
 whose diameter is equal to the diameter of the correspond- 
 ing sphere, and whose plane is perpendicular to the line 
 joining the centers of the two spheres ; then circle num- 
 ber one will cut off from each unit area of the circle number 
 two a definite fraction of all the particles which would other- 
 wise strike it from that side. This fraction would be the 
 quotient of the apparent area of the small circle as seen 
 
184 SPECULATIONS AND THEOBIES. [Part II. 
 
 from the unit area considered, divided by the total area of 
 the hemisphere whose radius is equal to the distance between 
 the two circles. The pressure per unit area of circle num- 
 ber two will, therefore, be less on that side by a definite 
 amount, which is proportional to the area of circle number 
 one ; hence the whole pressure on circle number two, being 
 the pressure per unit of area multiplied by the total area, 
 Avill be less than that on the other side by an amount which 
 is proportional to the product of the areas of the two circles. 
 This is only approximately true when the circles are of a 
 diameter which is not small compared to their distance 
 apart. 
 
 Considering any point on the circle number two, it will be 
 seen that the apparent area of circle number one, as viewed 
 from that point, will be inversely as the square of the dis- 
 tance between the circles. The fractional part of the total 
 pressure upon circle number two cut off by number one will, 
 therefore, be inversely as the square of the distance between 
 the circles. This is also only approximate, but more nearly 
 so in proportion as the distance between the circles is greater 
 in proportion to their diameter. 
 
 Replacing the circles now by the original spheres, it is 
 evident that the pressures will be the same as before, and 
 each sphere will therefore experience a force towards the 
 other which is proportional directly to the product of the 
 equatorial cross-sections of the spheres, and inversely as 
 the square of the distance between their centers. 
 
 If the spheres are not wholly impermeable, but are par- 
 tially permeable, then each will cut off from the other only 
 a portion of the streams of corpuscules. The law as to dis- 
 tance will clearly remain unchanged, but the amount of the 
 force will now depend not merely upon the equatorial cross- 
 section of the sphere, but upon the structure of the sphere. 
 If we imagine a sphere to be made up of separate particles 
 with considerable space between them, the particles them- 
 
Chap. III.] LE SAGE'S THEORY OF GRAVITATION. 185 
 
 selves being impervious to the corpuscules, then the amount 
 of the screening by the sphere as a whole will depend on 
 the disposition and size of its particles. Eventually, parti- 
 cles nearer the surface will tend to screen those below the 
 surface, so that the total screening will not be directly in 
 proportion to the total number of particles composing the 
 sphere and, therefore, to its mass. In order that all of the 
 particles shall be equally exposed to the bombardment of 
 the corpuscules, or nearly enough so, it is necessary to de- 
 vise some open structure for the material of the body. This 
 Le Sage did by imagining the particles of the sphere to be 
 not spherical or compact particles, but to have their material 
 distributed in lines along the edge of open cubes or other 
 geometrical figures. This structure gives to each particle a 
 greater surface exposure in proportion to its mass, and still 
 leaves a very open structure of the whole sphere, so that 
 the corpuscules may more easily penetrate it. By imagin- 
 ing this openwork structure carried to the necessary extent, 
 and by assigning to the corpuscules any necessary degree 
 of minuteness, the screening can be made as nearly as de- 
 sired in direct proportion to the mass of the sphere. It is 
 essential that the structxire should be so open and the cor- 
 puscules so minute that by far the greater part of them pass 
 freely through the material bodies, even when they are of 
 the size of the sun or other celestial masses ; because there 
 is no evidence that the action of gravitation between any 
 two such bodies is lessened when a third similar body is 
 interposed. This is shown on a large scale in the case of 
 the attraction between the sun and the moon at the time 
 when these and the earth lie nearly, but not quite, in the 
 same straight line as compared with the time when they lie 
 quite in the same straight line so that the moon is eclipsed. 
 Imagine, then, two such spheres, and we shall have the 
 forces upon each proportional to the jjroduct of the respec- 
 tive masses, and inversely as the square of the distance of 
 
186 SPECULATIONS AND THEORIES. [Part II. 
 
 their centers, as is demanded by the observed (Newtonian) 
 law of gravitation ; moreover, the approximation to this law 
 is no longer the rough one holding for the parallel circles, 
 but is a much closer one, depending only on the degree of 
 fineness of the corpuscules and on the openness of the struct- 
 ure of the sphere. The observed law of gravitation can be 
 also easily demonstrated for bodies other than spheres, these 
 having been taken merely for simplicity. 
 
 What has been shown for two spheres may be extended 
 to the mutual relation of any number of gravitating bodies, 
 and we then have a mechanical explanation of the cause of 
 the force of gravity substantially as given by Le Sage. 
 
 Objections. — In order that the Le Sage system of bom- 
 barding corpuscules should produce a resultant pressure as 
 supposed, when acting on two neighboring bodies, the cor- 
 puscules must give up all or a part of their translatory energy 
 at impact with the molecules of the bodies. Otherwise the 
 corpuscules rebounding from the near sides of the bodies will 
 exactly replace those screened out by the bodies, and there 
 will be no " shadow " or resultant force. The corpuscules, 
 being inelastic, cannot acquire vibration through impact, 
 neither can they receive rotation unless friction be presup- 
 posed, so that without loss of energy they cannot lose trans- 
 latory velocity, and, therefore, cannot cause gravitation. 
 But the force of gravitation to be explained exists not only 
 when the bodies are in approach but equally when they are 
 at rest or receding from each other. Thus it is requisite 
 that the corpuscules shall be yielding energy to the bodies at 
 all times. Also, as there can be no discrimination, this must 
 be true of every impinging corpuscule. What becomes of all 
 this immense quantity of energy beyond that part used in 
 such bodies as may be approaching one another ? And what 
 is the source of this energy ? These two queries indicate 
 difficulties which are fatal to the Le Sage theory in its origi- 
 nal form, or at least to the inelastic corpuscule which Le 
 
Chap. III.] LE SAGE'S THEOBY OF GRAVITATION. 187 
 
 Sage invokes. No theory is tenable which, to explain the 
 limited phenomena of gravitation, postulates a disproportion- 
 ate rate of expenditure of energy and one which must be 
 perpetually maintained from an unknown source. Add to 
 this that the stream of energy must be inevitably trans- 
 formed directly into heat, and one might suppose the theory 
 beyond hope of resuscitation. But good reason for its revi- 
 val seems to be supplied by the vortex-atom theory to be 
 developed. 
 
 We will here however review some obvious modifications 
 of the Le Sage theory. Gravitation seems to have to do 
 directly with the atom rather than with the molecule. All 
 considerations appear to impose perfect elasticity upon the 
 atom. Adopting this view, what would be the result of as- 
 suming the Le Sage corpuscules to be also perfectly elastic ? 
 If no vibration or rotation of the corpuscules resulted from 
 their impacts on the atoms, the system would be impotent to 
 produce gravitation. The case would be like the invisibility 
 of a body entirely enclosed by surroundings all at the same 
 temperature. There would be no corpuscular shadow be- 
 tween bodies. The sum of the transmitted and reflected 
 corpuscules would be the same in the same direction at all 
 points. But as the corpuscules and atoms are elastic, each 
 collision will be attended by a change in amount of energy 
 of vibration in both corpuscule and atom. Also if there is 
 any friction at the points of impact, or if the corpuscule is 
 not spherical, there will likewise be a change in energy of 
 rotation of the corpuscule. A Le Sage system of elastic cor- 
 puscules devoid initially of vibration and rotation would be 
 capable of producing the gravitation shadow. Por the re- 
 bounding corpuscules on the near sides of two bodies would 
 have less translatory velocity than the impinging corpus- 
 cules on the far sides, whose places they should make good 
 if the shadow is to be prevented. In other words, the aver- 
 age translatory energy of the corpuscules is less between 
 
188 SPECULATIONS AND THEORIES. [Part II. 
 
 two bodies than on the far sides, the ultimate reason being 
 the conversion of some linear into vibration or rotation 
 energy. At this point enters the objection raised by Lord 
 Kelvin but afterwards withdrawn : namely, that the trans- 
 latory energy of the system would be continually and rapidly 
 frittered away into distributed and unavailable energy of 
 vibration and rotation ; so that a continual and enormous 
 supply of energy would be needed to maintain the force of 
 gravitation even when no motion ensued. This otherwise 
 formidable objection is dispelled by Boltzmann's compensa- 
 tion proposition stated at page 177. This shows that in such 
 a system the vibration and rotation energies would each ap- 
 proach a definite ratio to the translatory ; and that if this 
 proportion were then in any way disturbed by external 
 action, the system would automatically proceed to restore 
 the balance. No energy therefore is consumed in the con- 
 tinuous operation of such a system except in the actual 
 performance of gravitation work. And the quantities de- 
 manded for the performance of even so enormous pieces of 
 work as the approach of the planets to the sun in their or- 
 bits is so utterly insignificant as compared with the whole 
 energy of the system of corpuscules, that its abstraction can 
 produce no sensible lessening of the force of gravitation. 
 
 A more serious difliculty arises from the consideration 
 of temperature equilibrium, as pointed out by Maxwell. 
 In the entire system of corpuscules and atoms, the condition 
 of temperature equilibrium is that the average kinetic 
 energy of the corpuscules shall be equal to that of the atoms. 
 Now in order to provide for the enormously long free path 
 of the corpuscules it would be natural to assume the num- 
 ber of them iu a unit of volume to be small, each one being 
 of the requisite mass. This would make the frequency 
 of collision smaller and the free path greater. But Max- 
 well asserted that under this condition the corpuscules 
 would raise the temperature of the atoms to white heat. 
 
Chap, in.] LE SAGE' S THEOBY OF GRAVITATION. 189 
 
 In other words, the kinetic energy of the corpuscules would 
 be far in excess of that of the atoms, so that bodies would 
 be inteusely heated. It has, however, been pointed out in 
 rejoinder that the assumption on which this objection rests 
 is apparently not an essential one, and that we are at 
 liberty to assume a large number of corpuscules per unit 
 volume if the temperature conditions can thereby be met ; 
 for the three variables, mass, number, and velocity, are 
 mutually adjustable to meet the facts which the hypothesis 
 has to explain. By increasing sufiiciently the number and 
 velocity of the corpuscules, and therefore diminishing their 
 size, there appears to be no intrinsic difficulty in providing 
 the necessary free path, and at the same time providing 
 temperature balance. At least, the contrary seems not yet 
 to have been shown. 
 
 It appears then that no objection has been advanced 
 which conclusively discredits the competence of a modified 
 Le Sage system to furnish a mechanical analogy to gravita- 
 tion. The elements and conditions of such a system would 
 be substantially these. Consider a space of stated dimen- 
 sions, throughout which gravitation is to be maintained. 
 Imagine a myriad of perfectly elastic particles or corpus- 
 cules moving through this space in such a manner that 
 all directions are equally represented in the motion, that 
 is, that a sensibly equal number of corpuscules will be 
 passing in every direction through a small sphere located 
 at any point of the space. The corpuscules must be exceed- 
 ingly minute as compared even with the atom of ordinary 
 bodies. Their mean velocity must be many times that of 
 light ; and their number per unit of volume, great as com- 
 pared with the number of atoms even in solids. The mean 
 free path of the corpuscules between collisions must be 
 many times the largest dimension of the prescribed space. 
 Finally, the system of corpuscules must be either of infinite 
 extent or surrounded by a perfectly elastic adiabatie boun- 
 
190 SPECULATIONS AND THEORIES. [Part II. 
 
 dary, that is, one whicli transmits no energy to or from the 
 system. The quantitative relation between the mass of 
 a corpuscule, the number per unit of volume, and the ve- 
 locity, is conditioned by the requirement of temperature 
 equilibrium between the system and material bodies. As 
 the temperature of all bodies is not the same, and as exact 
 equilibrium at one temperature will involve the heating of 
 bodies at a lower temperature, and the continual cooling 
 of those at a higher one, we discover a point of interest 
 which seems not to have been brought forward. To assign 
 to the corpuscular system the temperature of absolute zero 
 would involve a contradiction, since the system is based 
 on the kinetic theory of gases in which temperature is pro- 
 portional to the mean square of the velocity of the particles 
 in a given gas, and the corpuscules must of course be in 
 motion. On the other hand, space appears to be very cold, 
 so that the corpuscular temperature must be low. If then 
 gravitation is due to such a system, it miist be accompanied 
 by a continual cooling of nearly or quite all bodies. But 
 whether or not we adopt this last inference, we can hardly 
 escape the more general one, that some temperature must 
 be assigned to the system, and that bodies will therefore 
 be heated or cooled by the system according as they are 
 above or below that point. Have we here to do with an 
 insurmountable obstacle? There is certainly in operation 
 no known process of heat transfer with laws such as would 
 arise from this. The well-known thermal radiation into 
 space is a surface phenomenon, at least as contrasted with 
 this one which would proceed with equal facility at all 
 points throughout a body however large. This is suffi- 
 ciently serious, but on the other hand the fact must be recog- 
 nized that the whole phenomenon viewed as a case of the 
 interaction of two very diverse sets of elastic particles 
 specially conditioned, has not received adequate mathe- 
 matical investigation. Something not indicated in the 
 
Chap. III.] LE SAGE' S TUEOMT OF GRAVITATION . 191 
 
 kinetic theory of ordinary gases may be anticipated from 
 a rigid extension of that theory to meet the condition of 
 the joint system of corpuscules and atoms assumed to occur 
 within bodies. The enormous disparity between atomic 
 and corpuscular size, free path, velocity, and number per 
 unit of volume, form one ground for such anticipation. 
 Another is afforded by the entanglement of corpuscules 
 within the bodies with which they collide. This must 
 occur to a considerable extent because the gravitation 
 shadow is due to the transmutation of linear into rotational 
 or vibrational motion in the corpuscules, and further because 
 this change occurs not only in the relatively small portion of 
 the corpuscules engaged in producing a shadow, but in 
 every colliding corpuscule. Mathematical investigation 
 alone can clear up these points. 
 
 In the problem of temperature equilibrium, then, appears 
 to lie the crux of the Le Sage hypothesis. The forecast is 
 certainly not favorable, but there is still a future for the 
 theory. No other affords so comprehensive an analogy to 
 gravitation, and none is so helpful to the student who finds 
 his imagination unequal to coping with a purely abstract 
 notion of this fundamental property of bodies. 
 
 Two points in the theory call for further notice ; the first 
 of these is the permeability of bodies to the action of gravi- 
 tation. Le Sage was forced to meet this by a bold assump- 
 tion that the atoms are of an open structure as explained at 
 page 185. This, apparently not being demanded or justified 
 by other physical facts, has undoubtedly been a great in- 
 cubus to the theory. One cannot therefore refrain from a 
 feeling of pleasurable surprise at the manner in which the 
 vortex-atom removes this difficulty. To the hypothesis, 
 objection has also been raised on the score of the enor- 
 mously great velocities, great compared even with the 
 velocity of light, which it is necessary to ascribe to the 
 corpuscules ; and on the score of the extraordinary minute- 
 
192 SPECULATIONS AND THEORIES. [Pakt II. 
 
 « 
 ness necessary to be assigned to the corpuscules ; also of the 
 great length of free path between collisions. To these it 
 may be fairly answered that no objection upon the ground 
 of mere minuteness or immensity is really tenable. A 
 thing is merely great or small relatively to what we have 
 been accustomed to deal with, and there is no absolute scale 
 of magnitude. As to the length of free path, it is only nec- 
 essary that this should be assumed long as compared to the 
 dimensions of the solar system, or of such groups of celestial 
 bodies as double stars, etc., in which alone we have evidence 
 of the existence of gravitation. We are not called upon by 
 observed facts to account for any law of gravitation through 
 spaces greater than these, since the facts do not show that 
 such gravitation exists. 
 
 It is to be noted that the Le Sage bombardment theory 
 yields only an api^roximation to Newton's law of mass and 
 distance. For distances large relatively to atomic dimen- 
 sions, the approximation may be rendered as close as de- 
 sired by the assignment of proper corpuscular dimensions 
 and velocities. And this, it must be remembered, is all 
 that is called for by the facts. For our experimental 
 knowledge of the law is accurate only within limits, say 
 to one part in one million. Moreover, at short distances 
 there is no evidence that the force between molecules is 
 exactly that which would be indicated by Newton's law. 
 There is thus good reason to suppose that the law to be 
 accounted for may not be a constant one, but may change 
 with the distance. 
 
CHAPTER IV. 
 THE VORTEX-ATOM THEORY. 
 
 Vortex Motion. — This theory of the constitution of sub- 
 stances was suggested by Lord Kelvin soon after the publi- 
 cation in 1858 of Helmholtz's very remarkable mathematical 
 discussion of the possible rotational motion in a perfect fluid. 
 The theory has been developed from time to time both mathe- 
 matically and in its applications, particularly by Lord Kelvin 
 and Professor J. J. Thomson (cf. Helmholtz's Scientific Pa- 
 pers; Thomson, Sir William, Proc. Eoy. Soc. Edinb. 1867, 
 also " Nature," XXIV., 47 ; " Mathematical and Physical 
 Papers " ; J. J. Thomson's " Motion of Vortex Eings," Mac- 
 millan & Co., 1883 ; Maxwell, Encycl. Brit., article "Atom"). 
 The following sketch of this theory is designed to give 
 a general view of its scope and basis, not to present a 
 mathematical, exact, or complete statement of it, the latter 
 indeed being impossible without resort to a highly complex 
 mathematical treatment. 
 
 The theory undertakes to deduce by rigid mathematical 
 demonstration analogies to the properties and laws, both 
 physical and chemical, of material bodies, from the simple 
 laws of motion of an ideal perfect fluid. The purely mathe- 
 matical difficulties in the treatment of this subject have 
 restricted the progress of the hypothesis thus far chiefly to 
 the physical laws of gases, but a surprising extension of it 
 into the domain of chemical phenomena has been made by 
 J. J. Thomson. 
 
 It is essential first to define what is meant by a " perfect 
 fluid," and to examine the propertie.s ascribed to it. The 
 fluid whose motion is discussed is assumed to be "material" 
 o 193 
 
194 SPECULATIONS AND THEORIES. [Part II. 
 
 ill the sense that it possesses kiiiergety, the capacity for ki- 
 netic energy ; or some property from which kinergety may 
 arise. It is further assumed to possess inertia, to be contin- 
 uous in space, homogeneous, absolutely incompressible, and 
 frictionless (that is, devoid of internal friction). If a lim- 
 ited portion of the fluid is dealt with, it must be supposed to 
 be surrounded by a wall or boundary which is perfectly rigid, 
 so that whatever motion of the fluid takes place towards 
 the boundary, no displacement of that boundary occurs. If 
 we deal with an infinite volume of the fluid, no boundary 
 is requisite, because the infinite kinergety of the fluid with 
 reference to any point within the infinite volume is equiva- 
 lent thereto. No other attributes or conditions than these 
 are ascribed to the fluid in any part of the hypothesis or its 
 development. It is important to scrutinize these attributes 
 in detail. In doing so we are at once impressed by the fact 
 that they are all negative in character except kinergety ; 
 that is, that none of them requires the assumption of the 
 existence of energy of any kind, as would the property of 
 elasticity, or the assignment of forces to any portion of the 
 fluid or to its boundary. Thus inertia (cf. p. 19) is merely 
 the denial of any power of change of motion ; unless indeed 
 the term was employed to denote kinergety, — in which case 
 it brings no further attribute than has already been imposed 
 by that term. Inertia is properly only a denial of energy. 
 Homogeneity is merely the denial of any structural difEer- 
 ences in the fluid. Continuity is merely the denial of any 
 break in the fluid. It may be noted that in a finite mass 
 without a boundary, continuity could not be maintained, if 
 motion of any portion of the fluid took place, without the 
 existence of a force of cohesion or its equivalent, implying 
 energy ; hence, the necessity for the assumption of a boun- 
 dary or an infinite volume of fluid. Finally, absence of fric- 
 tion is merely the denial of the existence of any force of 
 friction, or any kind of action, within the fluid which would 
 
Chap. IV.] THE VOBTEX-ATOit THEOBY. 195 
 
 give rise to such a force when relative motion of parts of 
 the fluid occurred. In the section on "Resistance" it is 
 shown that frictional resistance of all kinds implies the 
 existence of some form of energy ; hence the denial of fric- 
 tion is merely another form of denial of the existence of 
 energy in the fluid. Incompressibility appears (at least 
 to the author) to be simply a denial both of elasticity 
 and of discontinuity; for it is not conceivable that the 
 total volume of a given portion of a substance can be les- 
 sened unless it consists of distinct portions with unoccupied 
 space between them, or unless the continuous material has 
 ascribed to it the property of elasticity. It may be re- 
 marked that this assertion of incompressibility is equivalent 
 to an assertion that matter occupies space, and leaves this 
 property of occupying space which is identical with incom- 
 pressibility, as a fundamental property of matter. 
 
 This hypothetical perfect fluid is, then, what this theory 
 presents as matter (or more properly as its analogue), using 
 this term in the sense in which it is defined in this book 
 and in which Lord Kelvin appears to employ it when he 
 chooses to discriminate between matter and material sub- 
 stances or bodies. 
 
 Any finite portion of an infinite volume of a perfect fluid 
 may have either or both of two modes of motion : namely, 
 first, translatory or irrotational motion ; second, rotational 
 or rotary motion. The second kind of motion exists in 
 any portion of the fluid when, if that portion be imagined 
 to be suddenly solidified, the solid would be in rotation 
 about some axis passing through it. The first kind of 
 motion occurs when any portion of the fluid is moving in 
 a manner such that if suddenly solidified the solid would 
 show no rotation about any axis. Lagrange, Euler, and 
 other mathematicians had investigated the irrotational 
 motion of such fluids, but the rotational motion had not 
 been discussed until Helmholtz published his paper on 
 
196 SPECULATIONS AND TBEOIUES. [Part II. 
 
 "Vortex Motion." In this he deduced the equations of 
 motion of rotation as well as of translation in a perfect fluid. 
 This was purely mathematical discussion with no reference 
 to the theory of the nature of matter. AMth the irrotational 
 motion we are not particularly concerned. 
 
 The discussion of the rotational motion led to very re- 
 markable and novel conclusions. 
 
 The first of these to be stated should perhaps be, that if 
 rotational motion has been imparted to any portion of the 
 fluid, that motion can never be transferred to any other 
 portion of the fluid, nor be changed into irrotational motion. 
 Thus any portion of the fluid once in rotation remains per- 
 manently in rotation. 
 
 It is also shown that the translatory motion of any por- 
 tion of the fluid cannot be transformed into rotational 
 motion ; this, however, had been previously demonstrated. 
 
 Thus rotational motion can never be produced or de- 
 stroyed in any portion of such a fluid and has therefore the 
 character of entire permanence. Energy of translatory 
 motion can be transferred from one portion to another but, 
 under the principle of the conservation of energy, cannot 
 be destroyed. 
 
 To picture the vortices, imagine a rubber rod of circular 
 cross-section with a diameter small compared with its 
 length and with its ends cut squarely off, to exist within 
 such a perfect fluid. That is, suppose for a moment that it 
 replaces its volume of the fluid. Imagine the rod to be in 
 rotation about its central line, that is, about a central axis 
 passing from one end to the other. Picture the motion of 
 each portion of such a rod as it revolves. Now imagine 
 that the rubber rod is replaced again by the fluid, and that 
 all portions of the fluid thus introduced move precisely as 
 the corresponding portion of the rubber rod previously 
 moved. We shall have here a circular column of fluid re- 
 volving upon a central longitudinal axis. Now Helmholtz 
 
Chap. IV.] THE VORTEX-ATOM THEORY. 197 
 
 showed that such motion can take place in only two cases : 
 first, where the column ends only at the boundary of the 
 liquid, that is, where each end extends to the boundary, 
 whether this be finite or infinite. Second, where the 
 column is of finite length, but where it has been bent 
 around until the opposite ends have been squarely abutted 
 against each other, thus forming a continuous ring instead 
 of a column. The motion within such a ring may be pict- 
 ured by imagining the rubber column as before in its 
 motion of rotation around its longitudinal axis, and then 
 imagining the column to be bent until the ends abut 
 squarely, and supposing those ends to be cemented together, 
 so that the ring becomes continuous. Each point in such a 
 ring will then be revolving about the central line of the ring 
 in a circular or nearly circular path whose plane is at right 
 angles to the central line ; and if we draw a radius of any 
 such circle, all the particles once on that radius will con- 
 tinue forever to remain upon it as the radius revolves with 
 the motion of the ring. In the supposed case, the ring will 
 be circular. If this rubber ring be now replaced by fluid 
 revolving in the same way, we have a revolving circular 
 ring of fluid which will always consist of the same parts 
 and will always retain its motion. Such a column, and 
 such a ring, of rotating fluid are called a vortex and a vortex 
 ring respectively. The central axis of rotation is called 
 the vortex line. In the case of the supposed ring this is 
 a circle whose diameter is the mean diameter of the whole 
 ring. Such a vortex ring once set in rotation is, therefore, 
 absolutely permanent in its existence and rotation, and by 
 the same reasoning, no such ring could be started by any 
 finite means, and could therefore exist only by "an act of 
 creative power." It is with the motion of such rings that 
 we are to be concerned, and not with the motion of the vor- 
 tex column. 
 
 The " strength " of a vortex ring is measured by the prod- 
 
198 SPECULATIONS AND THEORIES. [Part II. 
 
 uct of the cross-section of the ring into its angular velocity 
 of rotation. This product is shown to remain constant for 
 any given ring. Now as the total volume of such a ring 
 also remains constant, it is clear that if the ring enlarges 
 in diameter, and consequently diminishes in cross-section, 
 the angular velocity must increase in the same proportion. 
 The increase in diameter, therefore, is attended with in- 
 crease of angular velocity and diminution of cross-section ; 
 while a decrease of diameter is accompanied by the reverse 
 changes. 
 
 Vortex rings in the perfect fluid may possess translatory 
 motion and, therefore, the kinetic energy belonging to such 
 motion of matter. As there is no friction in the fluid, there 
 will be no retardation of the translatory motion of the ring 
 from that cause, and as the continuity of the fluid is main- 
 tained and it is incompressible, there will be no excess of 
 pressure upon the front of an advancing ring over the press- 
 ure upon the rear surface. That this should be the case 
 implies as stated above the existence of a rigid boundary 
 if a finite volume only of the fluid is dealt with, but if the ■ 
 volume is infinite, the infinite kinergety of the portion of 
 the fluid towards which the vortex is moving replaces the 
 effect of the boundary. 
 
 Kings similar in character to the vortex rings of a perfect 
 fluid may be easily produced in air, and will exist there for 
 sufficiently long periods for observation, although their 
 properties and duration in the air are affected by the fact 
 that this is not a perfectly frictionless fluid. Air rings may 
 be seen issuing from the smoke-stack of a locomotive under 
 certain conditions, being made evident either by the smoke 
 or condensed steam which is associated with them ; irregu- 
 larities in the distribution of the smoke or steam show 
 plainly the rotation in the cross-section of the ring. These 
 air rings also sometimes possess a rotation along the vortex 
 line around the center of the entire ring, a motion super- 
 
Ohap. IV.] THE VOBTMX-ATOM THEOBY. 199 
 
 posed upon the vortex motion proper. Similar rings may 
 easily be produced for experimental purposes by means of 
 a simple apparatus constructed as follows: a cubical box 
 eighteen inches on each edge has one side removed ; across 
 this is. stretched a piece of cloth or rubber, and in the oppo- 
 site side is made a circular opening a few inches in diame- 
 ter. If this box be filled with smoke, and a blow be sharply 
 struck upon the flexible cloth side, a puff of air carrying 
 smoke with it will be forced out of the circular opening, 
 and a circular vortex ring will move onward through the 
 air, its path and motion being made evident by the smoke. 
 Some of the properties of vortex rings can be exhibited by 
 means of these smoke rings. If two such boxes be prepared, 
 the rings discharged from one can be made to approach those 
 from the other, and they will be observed, when coming nearly 
 into contact with each other, to exhibit elasticity, rebounding 
 from one another with vibration precisely as we might imag- 
 ine rotating rubber rings to do under the same circumstances. 
 The peculiarities of motion when a smaller ring is made to 
 pass through, or approach, a larger one face to face, either 
 when both are moving in the same or opposite directions, 
 may be studied. If one box contains smoke and the other 
 only clear air, the effect of the action of the unseen ring upon 
 the visible one is said to be very striking. For rendering 
 the rings perceptible, it is convenient to employ instead of 
 ordinary smoke a cloud of ammonium chloride. If am- 
 monia vapor and the vapor from warm hydrochloric acid 
 are led through separate openings into one side of the box, 
 they will form a cloud of the white sal ammoniac dust in 
 the box. Further interesting experiments have been de- 
 scribed by Professor Ball ("Philosophical Magazine"). 
 
 It is shown that the vortex ring in a perfect fluid pos- 
 sesses an elasticity of form precisely analogous to that of 
 the smoke ring. If such a vortex ring be approached 
 obliquely by another ring, a change of form will take place 
 
200 SPECULATIONS AND THEOHIES. [Part II. 
 
 in both rings and they will rebound or glance off from one 
 another without coming to actual contact, and each after 
 rebounding will, in general, be in a state of more or less 
 violent, elastic vibration. This elasticity of form is due 
 wholly to the motion in the ring. It is, therefore, perma- 
 nent and perfect; and we have thus an object whose elas- 
 ticity of form is explained by the mere motion of a portion 
 of matter. This is the only means yet found of providing 
 a purely kinetic explanation for elasticity. In the kinetic 
 theory of gases, it will be remembered that in order to 
 explain by a kinetic method elasticity of the gas as a whole, 
 it is necessary to assume perfectly elastic particles or mole- 
 cules, or to ascribe to these molecules possession of forces. 
 Thus the ultimate explanation of elasticity was either 
 merely deferred or transferred to something equally diffi- 
 cult of explanation. 
 
 This property of elasticity renders vibration possible not 
 only through the change of form of the vortex ring as a 
 whole, but also through the change of form of the cross- 
 section of the ring. The periods of vibration will be de- 
 pendent upon the dimensions of the ring and upon its rate 
 of rotation. The possibility of such vibration at rates in- 
 herently dependent upon the ring itself is a point of utmost 
 importance in the development of the theory, particularly 
 in connection with the phenomena of radiation, a portion of 
 which constitutes the domain of spectroscopy. 
 
 We have thus far considered merely single, plane, circular 
 rings. There is however no reason why t-\^'0 such vortex 
 rings might not exist linked into one another, as are the 
 links of a chain ; or indeed a greater number of rings might 
 thus be linked, and the linkage might be of different degrees 
 of complexity. Any such linked set of vortex rings would 
 be a permanent structure incapable of separation of parts 
 by any process whatever ; for it is a fundamental property 
 of these rings that no ring can cut through the material of 
 
Chap. IV.] THE VORTEX-ATOM THEORY. 201 
 
 another, and that no ring can be in any way cut, since, as 
 Tait puts it, " it would wriggle away from the knife." We 
 have, thus, one means of producing units of differing degrees 
 of complexity. Again a ring may be imagined which shall 
 be knotted upon itself ; such as might be produced by start- 
 ing with our supposed rubber column, tying a knot in it of 
 any form whatever, of any degree of complexity, and finally 
 abutting and fastening together the free ends of the column. 
 The motion of rotation in such a true vortex ring would be 
 the motion which the parts of the rubber ring would possess 
 if the column were supposed to continue in the rotation 
 about its axis during and after the process of knotting and 
 splicing. Such a knotted ring is a mathematical possibility, 
 and its properties are determinate and are entirely analo- 
 gous to those of the plane circular ring. For reasons just 
 given, any such beknottedness is permanent ; thus we have a 
 second means of producing differences of character in rings. 
 Application to the Explanation of Matter. — The propositions 
 above stated comprise some of those originally deduced by 
 Helmholtz. Upon examining them, Lord Kelvin was at 
 once led to suggest that these vortex rings in a perfect 
 fluid afforded a new and most promising basis for a theory 
 of matter and of the constitution of material bodies. The 
 fundamental hypothesis as to the nature of the fluid is de- 
 void of the embarrassing assumptions attending that of solid 
 perfectly elastic or perfectly hard molecules. If we then 
 regard this perfect fluid as matter, and vortex rings in the 
 fluid as constituting the ultimate parts or atoms of bodies 
 or substances, we have the elements of Lord Kelvin's vortex- 
 atom hypothesis. The perfect fluid, although we have no 
 tangible evidence of its existence, demands of us the assign- 
 ment of only negative properties with the exception of kin- 
 ergety, and is the simplest conception yet put forward. 
 From this simple hypothesis, it remains necessary, in order 
 to shape the complete theory, to show by strictly mathe- 
 
202 SPECULATIONS AND THEOBIES. [Part II. 
 
 matical processes that bodies composed of such atoms 
 ■would have properties, aud would be governed by laws, 
 identical with those shown by experiment and observation 
 to belong to bodies coniposiug the physical universe. This 
 deduction is attended with great mathematical difficulties 
 and has on this account been carried to but a limited extent, 
 as already stated, but with results which are certainly most 
 remarkable. Before proceeding to consider some of these, 
 it may be well to note that this vortex ring possesses the 
 fundamental characteristics demanded of a material atom : 
 it is permanent in mass and volume ; it is indestructible ; it 
 possesses, through size, linkage, and beknottedness, the 
 possibility of varied but permanent individual character- 
 istics ; it possesses perfect elasticity without the necessity 
 of any assumption other than kinergety and motion. 
 
 Application to Gases. — Upon this hypothesis, a gas would 
 consist of a portion of space filled with the perfect fluid in 
 which were vortex rings of exceedingly minute diameter dis- 
 tributed at an average distance apart great as compared with 
 their diameter, and having random motion in all directions 
 with high velocities, probably somewhat as the molecules 
 of a gas in the ordinary kinetic theory. The following re- 
 marks will apply, unless otherwise stated, merely to such a 
 supposed gas. 
 
 The pressure of a gas consisting of vortex atoms is 
 explained in a manner closely analogous to that employed 
 in the familiar kinetic theory of gases, but of course with 
 modifications imposed by the characteristics of vortex-ring 
 motion. This explanation has been given by Professor 
 Thomson on the pages of " Nature '' above referred to, and 
 is substantially as follows : — 
 
 First, suppose the gas to be enclosed in a cylindrical , 
 vessel, with plane ends at right angles to the axis of the 
 cylinder. Imagine a vortex ring approaching one of these 
 ends with its plane parallel to the surface. As the ring 
 
Chap. IV.] THE VOUTEX-ATOM THEORY. 203 
 
 comes into contact with the surface, it gradually expands 
 and slowly spreads out over the surface until it reaches 
 the walls of the cylinder. It will then crawl slowly back- 
 wards along the sides of the cylinder towards the other end. 
 It will give lip momentuin during this contact with the 
 end first considered, and will acquire momentum again on 
 leaving that end, the two amounts being equal ; it will, 
 therefore, exert a pressure as it approaches and as it recedes 
 from the end. Suppose next that we consider an indefinitely 
 large and closed mass of the gas, and suppose a large num- 
 ber of vortex atoms constituting the gas to be moving about 
 in this enclosure. At any given instant, a number of these 
 will be approaching any surface of the boundary or any im- 
 aginary surface in the enclosure. Suppose the number thus 
 approaching to be large per unit area of the surface, then as 
 the rings approaching the surface gradually expand, they 
 will encounter portions of the surrounding atoms instead of 
 the sides of the cylinder, as in the case first supposed, and 
 will tend to move backwards, preventing each other's expan- 
 sion. There will be thus produced a more or less thick 
 layer of tangled riags immediately upon and over the sur- 
 face, and at the end of a certain time after the beginning of 
 such an operation, a state of kinetic equilibrium will come 
 about in which as many rings will be coming away from 
 this layer with their original average diameters and ve- 
 locities, as will be entering into it. The gas will thus be 
 exerting a continuous and sensibly uniform pressure upon 
 the surface. The condition of this layer is complicated 
 by rings- approaching it more or less obliquely, but it is not 
 difficult to see, in a general way, that the resulting effect of 
 this action is essentially the same as that of the bombard- 
 ment by elastic spheres described in the kinetic theory of 
 gases. This condensed layer at the boundary between a 
 solid and a gas must play an important part in a discus- 
 sion of the observed phenomena of surface condensation. 
 
204 SPECULATIONS AND THEORIES. [Pakt II. 
 
 By a brief mathematical demonstration, Boyle's law of 
 the compressibility of gases can be shown to hold for a sub- 
 stance thus constituted. The equation which represents 
 this law contains a small term which shows a deviation from 
 the law on the side of greater compressibility. This so far 
 as it goes accounts for the observed deviation of ordinary 
 gases from the law at ordinary pressures. It is to be noted 
 that this term involves no new fundamental assump- 
 tions. 
 
 The relation between the kinetic energy of vortex rings 
 and the absolute temperature of the gases composed of them 
 as just described, appears not to have been worked out with 
 any fulness. For a gas consisting of single vortex rings, 
 the square of the velocity is not proportional to the temper- 
 ature. As the temperature rises, the diameter of the ring 
 increases and its velocity lessens, although the total momen- 
 tum of the atom increases with the temperature. For such 
 a gas, therefore, it appears that the temperature would be as 
 some inverse function of the velocity rather than in propor- 
 tion to it. With more complicated ring atoms, a different re- 
 lation is indicated between temperature and velocity. These 
 facts constitute a point of radical difference between the 
 former kinetic theory of gases and the vortex-atom theory, 
 and J. J. Thomson has suggested that upon this difference 
 might be based a crucial test of the two theories. The phe- 
 nomena of thermal effusion of gases is pointed out as one of 
 those whose laws, as deduced from the two hypotheses, 
 should be radically different. A test would thus be afforded 
 by the comparison of the deductions from the respective hy- 
 potheses, with experimental observations, under correspond- 
 ing conditions. jSTo such tests have yet been carried out, and 
 they are likely to be attended by serious difficulty, both 
 mathematical and experimental. The fact that temperature 
 is not proportional to the square of the velocity of the vortex 
 atom will probably lead to a different condition for tempera- 
 
Chap. IV.] TBE VORTEX-ATOM THEORY. 205 
 
 ture equilibrium between different gases and between sub- 
 stances in general, from that pointed out in the kinetic 
 theory of gases, namely, the quality of kinetic energy of the 
 molecules. If so, this would probably have a very impor- 
 tant bearing upon many applications of the hypothesis, 
 notably in the explanation of gravitation, which will be 
 later referred to. 
 
 Applications in Chemistry. — The extension of the vortex- 
 atom hypothesis to the explanation of various chemical laws 
 and phenomena has been chieiiy made by Professor J. J. 
 Thomson, and is mainly restricted to gases. 
 
 The starting-point is a system of vortex rings in which 
 every individual ring has the same, or very nearly the same, 
 strength whether it is a single ring or one which is in any 
 way associated with other rings. In other words, the fun- 
 damental unit is the single ring always of a given strength. 
 It is to be remembered that the volume and strength and kin- 
 ergety of such a ring are invariable, but that the diameter of 
 the ring varies with sundry conditions, such as temperature, 
 etc., necessitating a corresponding change in either angu- 
 lar velocity or cross-section, in order to maintain constant 
 strength. The further condition is imposed that the cross- 
 section of the ring is always exceedingly small in propor- 
 tion to the diameter of the ring. 
 
 Single rings of equal strength may differ from each other 
 not only as thus shown, but may be intrinsically different 
 in the following ways. Different rings may possess differ- 
 ent volumes, and therefore different kinergeties, although of 
 equal strength. This may come either through differences 
 of cross-section or through beknottedness. The latter condi- 
 tion may naturally be supposed to account for characteristic 
 differences other than those arising from mere differences 
 of kinergety. 
 
 Besides the difference between single rings, further differ- 
 ences in the character of bodies may come about in either 
 
206 SPECULATIONS AND THEOlilES. [Part 11. 
 
 of two ways. Two or more rings may be linted into one 
 another ; two or more rings may be associated together by 
 rotation about one another. These two methods of combi- 
 nation afford the basis of a theory of chemical action, and 
 will now be explained in more detail. They serve, with 
 no additions to the fundamental hypothesis, to give very 
 instructive explanations of chemical affinity, atomicity, va- 
 lency, dissociation, and other chemical laws. The theoreti- 
 cal explanation of the constitution of the atom and of atom- 
 icity is based on the linkage of vortex rings. The simplest 
 possible atom is supposed to consist of a single vortex ring 
 of the most elemental form. This constitutes a monad atom, 
 but of course monads may differ in characteristics, through 
 differences in kiuergety and beknottedness as just described. 
 
 More complicated atoms are supposed to consist of two 
 or more single rings linked into one another. We have then 
 to consider next the system of linkage and the resulting 
 atoms. 
 
 We may imagine two or more rings linked together in a 
 great variety of ways. Probably many of the systems thus 
 produced will prove, on mathematical analysis, to be un- 
 stable assemblages. They could therefore not be supposed 
 to be originally formed, or if formed they must either re- 
 arrange themselves into some stable position, or break up 
 into other and stable groups. The simplest method of link- 
 age which has thus far been shown to be stable, and which 
 upon investigation seems to afford all the necessary com- 
 plexity, is as follows. The statements will be made only 
 for rings of equal strength, equal diameters (under the same 
 condition), of a cross-section small compared with the diame- 
 ter of the whole ring, and where the shortest distance be- 
 tween combined rings is small compared with the cross- 
 section of the ring, and rings without knots. The extension 
 to other cases has already been sufficiently indicated for the 
 purpose in hand. 
 
Chap. IV.] THE VOETEX-ATOM TBEOBY. 207 
 
 Suppose the rubber column above described (p. 196) to 
 have, traced upon its surface, a spiral line like a screw thread, 
 passing once or more times completely around the col- 
 umn in going from one end to the other. If, then, the col- 
 umn be bent around into a ring with its ends abutted, the 
 two ends of this line will come together, and we shall have 
 a spiral line around a circular ring. Imagine that line to 
 be a vortex ring of small cross-section. We shall thus have 
 a spiral vortex lying upon the surface of a circular ring. 
 Again suppose the column to be straight, and that two such 
 spiral lines be drawn about it everywhere equidistant from 
 one another. This will be like a two-threaded screw. Sup- 
 pose the spirals each to make a whole number of turns in 
 passing from one end of the column to the other. Imagine 
 the column then bent around into a circle and the ends 
 abutted. One end of each spiral will thus meet with the 
 other end of the same spiral. If, then, we imagine each of 
 these spirals to be vortex rings, we shall have two spiral 
 vortex rings lying on the surface of a circular ring. Sup- 
 pose the rubber now to disappear and to be replaced by non- 
 rotating fluid. We shall evidently have two spiral vortex 
 rings linked into one another in this special manner. The 
 rings will be everywhere equidistant from one another as 
 measured by the shortest line between them, and it may be 
 shown mathematically that such a pair of linked rings is a 
 stable arrangement. That is, if any part of either vortex 
 be slightly displaced, the system will tend to return to its 
 initial condition. This is the supposed constitution of a 
 dyad atom. 
 
 Instead of two spiral rings corresponding to a two-threaded 
 screw, we may have three, four, five, or six, arranged in pre- 
 cisely the same way. Any of these systems of linked vor- 
 tices will be stable, but any system of seven or more vortices 
 is shown to be unstable, a fact of great importance in con- 
 nection with chemical theory. The triad, tetrad, pentad, 
 
208 SPECULATIuys and theories. [Part II. 
 
 and hexad atoms are supposed to be thus constituted ; and 
 under this hypothesis no higher atomicity is possible. 
 
 This system of linkage may be slightly modified as fol- 
 lows. Suppose, in drawing two spiral lines around the 
 rubber column, that instead of making a whole number of 
 turns, each line makes an extra half-turn about the column. 
 Then, when the ends of the column are abutted, the begin- 
 ning of one spiral line will join the end of the other spiral 
 line and vice versa; so that if now we start at any point, 
 upon either of the lines, and follow it over the ring, we 
 shall find that the two spirals are continuous with one 
 another. That is, instead of having two separate linked 
 spirals, we have one continuous spiral looped into itself. 
 The properties of such a spiral, or rather of the correspond- 
 ing looped vortex ring, are shown to be substantially the 
 same as of the two spiral linked vortices. Similar descrip- 
 tion would show how we may have continuous linked vor- 
 tices corresponding to three, four, five, or six separate linked 
 vortices. 
 
 It is not denied that there may be other methods of link- 
 age which will be stable, but no others are obvious and 
 none have yet been discussed with any considerable ful- 
 ness. As has already been noted, any one or more of the 
 component spirals in the complex vortices may be in itself 
 complex. 
 
 The theoretical explanation of chemical combination, 
 that is, the union of atoms to form molecules, either ele- 
 mentary or compound, is based upon the association of 
 rings through the process of rotation about one another. 
 
 If two rings approach one another in any direction and 
 in any relative position, they will in general collide or come 
 nearly into collision and will rebound from one another. 
 The respective velocities and direction of motion will be 
 changed by the collision, and more or less change in their 
 rates and periods of vibration will occur. But under cer- 
 
CiiAp. IV.] THE VORTEX-ATOM THEORY. 209 
 
 tain circumstances, the two rings will not separate but will 
 remain associated in rotation about one another. The fol- 
 lowing is an illustration. 
 
 Consider two simple, circular, vortex rings. If they ap- 
 proach each other, moving in the same direction, with their 
 planes parallel or nearly so, and their centers very nearly 
 in the same line perpendicular to their planes, then if the 
 rear ring overtakes the forward one, it will remain associ- 
 ated with it instead of passing away from it, under certain 
 conditions. These conditions are that the strength of the 
 two rings shall be very nearly equal, and that the difference 
 in diameter of the two shall be small in proportion to the 
 diameter of either. The two rings thus meeting will con- 
 tinue to revolve about one another in a way which may be 
 thus described. Suppose the two rings to be of exactly 
 equal strength ; to have their planes parallel ; and suppose 
 that the rear ring is slightly smaller than the forward ring. 
 When they reach the same plane, suppose their centers to 
 coincide, and a circle to be drawn midway between the two 
 rings. Instead of separating, the rings will remain associ- 
 ated, revolving about this circle, until some external disturb- 
 ing cause forces them asunder. If the rings are nearly but 
 not exactly concentric, and nearly but not exactly of equal 
 strengths, they will similarly remain associated, revolving 
 about a circle lying between them, and located so that its 
 distance from each is inversely proportional to the strength 
 of the ring. The shortest perpendicular distance between 
 the rings will remain constant and the pair of rings will 
 continue to move forward. It is only when the diameters 
 are within a certain limit of each other that the rings are 
 capable of remaining associated at all. The more nearly 
 equal the strengths of the rings, and the more nearly equal 
 the two diameters, the greater will be the disturbance requi- 
 site to cause their separation. We have thus the explana- 
 tion of chemical combination between two atoms, either of 
 
210 SPECULATIONS AND THEORIES. [I'art II, 
 
 the same or of different kinds ; also a clew to the reason un- 
 derlying the relative stability of such compounds. 
 
 If three or more rings are brought together at once, or in 
 succession, in a manner analogous to that just described for 
 two rings, they will all remain together and revolve about 
 a common line in a manner similar to the two rings, pro- 
 vided that similar conditions are fulfilled. But the chance 
 of fulfilment of such a combination of conditions is much 
 smaller for an assemblage of three rings than for the com- 
 bination of two rings. Also for equal conditions, the stabil- 
 ity of the combination is less ; so that we should expect to 
 find very many more molecules of two rings than of three, 
 or four, or more. 
 
 Each of these two or more rings thus associated with each 
 other to form a molecule may be called a primary of the 
 molecule. These primaries, in order that the molecule 
 should be permanent, must be of equal or very nearly 
 equal strengths. It is very important to fix clearly in 
 mind two points : no molecule can be stable whose prima- 
 ries are not very nearly equal in strength, and no molecule 
 composed of more than six primaries will be stable. 
 
 Any primary may be either simple or complex, but if it 
 is complex it consists of two or more associated rings. In 
 this case each of the rings associated to form the primary 
 is called a secondary. These secondaries may be associated 
 in either of two manners. First, there may be two or more 
 independent rings rotating about each other in the manner 
 described, and thus constituting a molecule. Second, there 
 may be two or more rings linked together as above detailed, 
 in which case the secondaries are permanently associated 
 and therefore constitute more or less complex atoms. 
 
 Chemical combination may be regarded somewhat in this 
 light. Suppose a limited space to contain a large number 
 of atoms of all degrees of complexity, made up as above de- 
 scribed. These would then represent atoms of all elements; 
 
Chap. IV.] THE VORTEX-ATOM TEEOBY. 211 
 
 and it may be here noted that while we have but seventy 
 diiferent kinds of atoms to be accounted for, there is an 
 almost infinite range of possibility in the complexity of 
 vortices. At the same time the atomicity of the vortex 
 atom is limited to six, — a deduction which strikingly tallies 
 with chemical facts. Suppose the atoms to be small com- 
 pared with the total volume of the space, and that they are 
 all moving freely about in all directions. In a multitude 
 of collisions which will take place, the conditions necessary 
 for the combination of rings above indicated will be fre- 
 quently fulfilled. That is, two atoms of equal strengths 
 and nearly equal diameters will come into collision in such 
 a manner as to remain rotating about one another. If we 
 note that, as already stated, the unit, single ring in the en- 
 tire system is of the same strength, and that the strengths 
 of linked rings is the sum of the strengths of the component 
 rings, we shall see that the dyad atoms have double the 
 strength of the monad atoms, and so on. Thus the simplest 
 combinations would be pairs of monads, pairs of dyads, and 
 so on. Thus the first molecules to be formed, and the most 
 stable ones, would be diatomic molecules of monad elements. 
 Of these the most stable compounds would apparently be 
 those whose atomic strengths were the most nearly equal, 
 and whose atoms were the simplest. Combinations of two 
 dyad atoms to form a molecule would probably be less fre- 
 quent and less stable, since for these the conditions are 
 slightly more complex. A similar line of remarks applies 
 to the triads, and so on. Combinations of three or more 
 monad atoms to form a molecule would be much less fre- 
 quent, and similarly of three or more atoms of higher 
 atomicity. 
 
 A single monad atom could not form a stable compound 
 with a dyad atom or with any atom or molecule more com- 
 plex than a single monad. 
 
 A dyad atom may combine with any other dyad atom. 
 
212 SPECULATIONS AND THEORIES. [Part II. 
 
 This ■would constitute a molecule having two primaries, 
 each primary being a dyad, that is, having two linked 
 vortex rings. 
 
 Or a dyad atom may combine with a primary consisting 
 of two monad atoms associated in rotation; that is, the 
 dyad maj'' combine with a molecule consisting of two 
 monad atoms. It cannot combine alone with an atom of 
 higher atomicity or with a primary of more than two 
 monads, on account of inequality of strength. 
 
 Or a dyad may combine with two other primaries, which 
 may be, respectively, both dyad atoms ; both, molecules 
 consisting of t^^o monads ; or one, a dyad atom, the other 
 a molecule of two monads. This molecule of three prima- 
 ries has less stability than the preceding one of two prima- 
 ries. 
 
 We may proceed in a similar manner, under the condition 
 that all primaries must be of equal strength, to demon- 
 strate what atomic combinations into molecules are possible 
 or impossible. For instance, a triad atom could not form 
 a stable molecule with a dyad atom or with a dyad atom 
 and a monad atom. The tetrad atom could not combine 
 with a triad atom, nor with a triad and a monad, nor even 
 with two dyads, unless they were associated as a pair in 
 rotation. 
 
 These illustrations will suffice to show how readily the 
 hypothesis accounts for atomicity, quantivalence, and chemi- 
 cal combination in general. We may however note that 
 it has not yet accounted for the changes of energy attend- 
 ing chemical actions, which shows itself so largely in the 
 production or absorption of heat and electrical energy. 
 We will now apply the hypothesis to certain typical chemi- 
 cal combinations. From the usual considerations and some 
 others based upon vapor-density determinations and so on. 
 Professor Thomson, in his book above referred to, has 
 classified the elements provisionally according to their 
 
Chap. IV.] THE VORTEX-ATOM THEORY. 213 
 
 atomicity, valency, or qnantivalence. His classification 
 is only partial and is not to be regarded as a final one, not 
 only because of the diffictilty attending such a classification 
 ordinarily, but from certain added ones which the novelty 
 of this hypothesis naturally induced. According to his 
 grouping : — • 
 
 The monads or univalent elements consist of : — 
 
 Arsenic, Bromine, Chlorine, Fluorine, Hydrogen, Iodine, 
 Mercury, Nitrogen, Phosphorus, Potassium, Rubidium, and 
 Thallium. 
 
 The dyads or divalent elements are : — ■ 
 
 Cadmium, Carbon, Chromium, Copper, Lead, Manganese, 
 ]\rercury. Oxygen, Selenium, Sulphur, Tellurium, Zinc. 
 
 The triads or trivalent elements are : — 
 
 Aluminum, Antimony, Bismuth, Boron, Indium. 
 
 The tetrads or quadrivalent elements are : — 
 
 Silicon, Tin. 
 
 Thus the atom of hydrogen consists of a single vortex 
 ring ; the atom of oxygen of two linked vortex rings ; the 
 atom of aluminum of three linked rings ; the atom of sili- 
 con of four linked vortex rings. 
 
 The molecule of hydrogen consists of two hydrogen atoms 
 rotating around one another. The molecule of oxygen con- 
 sists of two atoms of oxygen rotating around one another, 
 that is of two pairs of linked vortices, each pair rotating 
 around the other, and so on. 
 
 The molecule of hydrochloric acid, HCL, consists of two 
 primaries associated in rotation, each primary being a single 
 vortex ring. The molecule of water, OHj, consists of two 
 primaries associated in rotation. The first primary is an atom 
 of oxygen, namely, two linked vortex rings. The second is 
 a molecule of hydrogen, that is, two single vortex rings as- 
 sociated in rotation. 
 
 "The sulphur compounds afford good examples of mole- 
 cules containing various numbers of primaries, that is we 
 
214 SPECULATIONS AND THEORIES. [Pari II. 
 
 have HaS with two primaries Hj-S; SO2 with three pri- 
 maries S-0-0 ; and SO3 with four primaries S-0-0-0 " (J. J. 
 Thomson, p. 122). 
 
 For an interesting discussion of the valency of nitrogen 
 and of carbon, reference must be made to the fourth section 
 of Thomson's book. 
 
 Dissociation. — It has already been stated that two rings 
 will remain associated together in rotation only when their 
 diameters are very nearly equal and their strengths also 
 nearly the same. The degree of stability therefore of differ- 
 ent molecules of the same substance will depend upon the 
 nearness to equality of diameters of the component atoms 
 in the given molecule, and the nearness to equality in the 
 strength of these atoms. The degree of stability of differ- 
 ent substances will depend upon similar conditions respect- 
 ing their primaries, and also upon further conditions. Of 
 these, the complexity of the molecule will be one; and 
 others would arise, it would seem, from difference in beknot- 
 tedness in the primaries or their components, and their dif- 
 ferences in kinergety, and possibility of angular velocity of 
 the primaries. 
 
 In a quantity of any given gas, the violence of the colli- 
 sion of molecules at any instant would be very diverse. It 
 is not unreasonable to suppose that undfer any stated condi- 
 tion of temperature and pressure, some of these collisions 
 would be sufficiently violent to split up the molecule. This 
 would require merely that the impact upon one or more ■ 
 atoms of the molecule should be so severe as to change 
 either its diameter or its velocity to a sufficient extent to 
 cause it to move away from the other atoms with which it 
 had been associated in the molecule. The component atoms 
 thus separated from each other would remain separated until 
 they came again into collision under the conditions neces- 
 sary for reunion, or until they met with others with which 
 they could combine. There would thus be throughout the 
 
Chap. IV.] THE VORTEX-ATOM THEORY. 215 
 
 gas, under a stated condition, a certain definite proportion 
 of atoms of disrupted molecules. This mechanical breaking 
 up of molecules is called dissociation, and a gas in the con- 
 dition just described is said to be partially dissociated. 
 
 The proportion of disrupted atoms would depend upon 
 the average stability of the molecules of the gas, and upon 
 the frequency of the collisions of great violence. Thus for 
 different gases, the degree of dissociation at a stated tem- 
 perature would depend upon the nature of the gas. For the 
 same gas the degree of dissociation would increase with the 
 temperature, and might also be to some extent a function of 
 pressure. At a sufficiently high temperature, all or a suffi- 
 cient number of the collisions might be so violent as to 
 maintain the whole or nearly all of the gas in a state of 
 dissociation. This condition would however be somewhat 
 gradually rather than abruptly reached, as the temperature 
 increased. These deductions from the theory tally very 
 closely with the fundamental facts of dissociation. It 
 would be interesting to see the theory further applied in 
 connection with the modern extension of the dissociation idea 
 of electrolysis and other portions of physical chemistry. 
 
 It is easily seen that a consideration of the conditions for 
 the formation of molecules from atoms, and for the stability 
 of the molecule when formed, must afford explanations of 
 other observed chemical phenomena. Thomson briefly dis- 
 cusses the application of the theory to the time-rates of for- 
 mation of chemical compounds, and to the fact that a given 
 compound, once formed, is stable over a greater range of 
 temperature than that over which its components can be 
 made to imite directly to form the compound. 
 
 Gravitation. — The vortex-atom theory has been applied 
 by Lord Kelvin to the explanation of gravitation, by point- 
 ing out its adaptability to the Le Sage theory. The possi- 
 bilities of this. theory when a perfectly elastic corpuscule 
 is substituted for Le Sage's inelastic and perfectly hard one. 
 
216 SPECULATIONS AND TIIEOUIES. [Part II. 
 
 have been indicated at page 187. A system of vortex rings 
 of the requisite fineness would fulfil all the apparent func- 
 tions of the system of corpuscules, besides affording a fun- 
 damental explanation of their elasticity instead of leaving 
 this in the position of an arbitrary postulate. Further, the 
 vortex atom incidentally provides substances with precisely 
 that openness of structure which Le Sage found essential to 
 his theory but which seemed an otherwise uncalled-for and 
 a rather fanciful suggestion. 
 
 The vortex ring then seems to lend itself to this theory of 
 gravitation, which far exceeds in merit, any other yet ad- 
 vanced, and to which the most serious difficulty lies in the 
 question of temperature equilibrium. The nature of this 
 objection has been presented at page 188 for the case of 
 elastic corpuscules having the character assumed for the 
 molecule in the kinetic theory of gases. For the vortex 
 corpuscule, somewhat the same objection awaits removal. 
 Briefly stated, it is this : the corpuscular and atomic sys- 
 tems can, presumably, be in temperature equilibrium at only 
 one temperature. The temperature of the corpuscular sys- 
 tem must be constant. Hence there must be heating of 
 some bodies and cooling of others by gravitation, which 
 seems contrary to fact. But this objection, although pre- 
 sumptively good, has not been rigidly proved. As already 
 remarked (cf. p. 189 ) the conditions of equilibrium under 
 the peculiar circumstances of the case have not been duly 
 investigated. Also there remain to be ascertained the conse- 
 quences of the change of vortex diameter with temperature, 
 and of the still unknown relation between vortex velocity 
 and temperature. The theory is in a condition far from 
 hopeless. Indeed, it appears to have arrived at that espe- 
 cially valuable stage, where further inquiry is definitely 
 outlined, which if carried out is sure to be helpful whether 
 it overturns or supports the whole fabric. 
 
 A consequence of the Le Sage theory, which appears to 
 
Chap. IV.] THE VORTEX-ATOM TUEOBY. 217 
 
 have escaped notice, is that the weight of the atom cannot 
 be proportional to its volume or mass ; since the Le Sage 
 corpuscules can act on the surface only of the atom. If 
 then we apply this remark to the vortex atom, noting that 
 the atoms are of the same average size, we see that the 
 gravitational force between two atoms should be propor- 
 tional to the product of their respective thicknesses. But, 
 as their diameters are the same, their volumes are propor- 
 tional to the sqiiares of the thickness of the ring. Hence 
 the gravitation force between two atoms must be propor- 
 tional to the square root of the volumes of the atoms. If 
 then we assume the fluid to be matter, then the weight, and 
 therefore the kinergety of an atom is proportional to the 
 square root of its mass. From this it would follow that 
 mass is proportional to the square of kinergety, and not to 
 kinergety directly as is ordinarily assumed (cf. p. 160). 
 
 Cohesion. — The force of cohesion in bodies is sensible 
 only when the portions involved are apparently in exceed- 
 ingly close proximity to one another. It is usually assumed 
 to be a force which increases much more rapidly than the 
 gravitation law of inverse squares would account for. This 
 may however be only apparent, for if the atoms and mole- 
 cules of bodies are composed of vortex rings, then as they 
 approach one another, portions of the rings would be in con- 
 tact or very near to one another, on the average, while the 
 centers of the rings were still at relatively long distances. 
 If, then, gravitation is due to a bombardment of the kind 
 above supposed, the force holding two rings together when 
 in contact at one point only, or nearly so, would be very 
 much greater than the force of gravitation calculated from 
 the ordinary law upon the basis of the distance between the 
 centers of the rings. In other words, the distance law of 
 gravitation upon this hypothesis may cease to hold at dis- 
 tances which are not many times the dimensions of atoms 
 or molecules. It is quite possible, therefore, that the force 
 
218 SPECULATIONS AND THMORIES. [Pari 11. 
 
 of cohesion may be due to the same action which produces 
 gravitation ; in other words, that the energy causing cohe- 
 sion is identical with that causing gravitation. 
 
 Elasticity. — Tlie elasticity of form of the vortex rings, 
 arising solely from their energy of rotation, seems to afford 
 a quite sufficient basis, in connection with the theory of 
 cohesion, to account for the observed phenomena of elas- 
 ticity in material bodies. Some possible applications of 
 the theory to the explanation of these phenomena will be 
 briefly suggested in a manner seemingly consistent with 
 the usual presentation of the theory. 
 
 A single vortex ring when stretched along a diameter 
 will have its transverse diameter shortened. It will oppose 
 to the stretching force a resisting force proportionate to the 
 amount of the distortion, if this be small. A similar state- 
 ment holds for compression of a ring. A straight row of 
 such rings, in contact with each other at the ends of diam- 
 eters, and exposed to a compressive force acting along the 
 row from each end, would exert an opposing elastic force 
 proportional to the amount of compression. If such a row 
 of rings were fastened together at their points of contact, 
 and if each end ring were acted upon by a force tending to 
 pull the rings apart in the direction of the row, the row 
 would stretch by the distortion of the individual rings and 
 would exert an elastic resisting force proportionate to the 
 amount of stretch. The row would break apart when the 
 stretching force exceeded the force holding together any 
 two successive rings — the part played by the force of cohe- 
 sion in material bodies. It is easy to see that a body made 
 up of an aggregation of such rows of rings, and having an 
 equal number of rows in every direction, would have some, 
 at least, of the elastic properties of an isotropic solid. And 
 similarly we may see that a body composed of vortex rings 
 whose axes were scattered equally in all directions at ran- 
 dom would likewise have some of the elastic properties of 
 solids. 
 
Uhap. IV.] THE VORTEX-ATOM THEORY. 219 
 
 If the force holding the rings together, that is, the force 
 of cohesion, were caused by a form of energy of the nature 
 of that above ascribed to gravitation under the vortex the- 
 ory, then the cohesive force would be insignificant except 
 at the points of actual contact. It would diminish very 
 rapidly in amount as- the contact points were moved apart. 
 Hence a force sufficient to overcome cohesion vi^hen the 
 rings were actually in contact would be much more than 
 sufficient to continually increase their distance, after sepa^ 
 ration. Whether a body composed of rings, not in actual 
 contact with their neighbors, would exhibit the character- 
 istics of a solid (e.g. rigidity) may be difficult to say. In 
 such a body, the separation of the rings would presumably 
 be maintained by their translatory motion and collisions. 
 The effect of cohesive force in any such body would appar- 
 ently be very small. The body would show little or no 
 elastic resistance to extension, and its resistance to com- 
 pression would be less than that of the same body with its 
 rings in contact — very much less when the average distance 
 between the rings became large in comparison with the diam- 
 eter of the rings. 
 
 If, as it appears, the diameter of the vortex ring in- 
 creases with the temperature, then the most compact ag- 
 gregate of rings at a given temperature would still be 
 capable of contraction with diminution of temperature, 
 although capable of little or no reduction of volume by 
 increase of pressure. 
 
 Remarks. — The very important chemical explanations 
 arising as a direct result of this vortex theory, the remark- 
 able extension of the applications of the theory already 
 made, and their concordance with observed facts, indicate 
 for this hypothesis far greater promise than attends the 
 former, kinetic, theory or any of the other suggestions yet 
 made. Its slow growth is clearly due to the rare skill re- 
 quired to deal competently with the mathematical problems 
 
220 SPECULATIONS AND THEORIES. [1'art II. 
 
 that it presents. If tlie theory has won less general ac- 
 ceptance than the careful study of it might seem to warrant, 
 this is doubtless to be ascribed to the inevitable complexity 
 of its mathematical treatment, and the want of an expo- 
 ■ sition which should clearly state the principles and results 
 of the hypothesis in a manner intelligible to those not 
 familiar with the higher branches of mathematics, or with- 
 out the leisure to review the extensive literature of the 
 subject. At a first glance, the fundamental conceptions 
 involved in the theory may appear fanciful; but looking 
 below the surface, we see, as has above been indicated, that 
 this is distinctly contrary to the fact, since the properties 
 assigned to the fluid imagined are of the simplest possible 
 sort and with the exception of kinergety are negative in 
 character. That no such fluid is known to us through any 
 other means constitutes no objection to the hypothesis. 
 There is no obvious reason why, if such a fluid does exist, 
 we should be in any way aware of it except when it is in 
 vortex motion. The fact that the motion of the rings can- 
 not be philosophically accounted for is clearly not an objec- 
 tion, for it is no part of physics to account for the origin 
 of matter or motion. 
 
 The following quotation gives the greater part of Max- 
 well's comment on the vortex-atom theory. It is taken 
 from his article "Atom " in the Encyclopedia Britannica: — 
 
 " On the other hand, the vortex ring of Helmholtz, im- 
 agined as the true form of the atom by Thomson, satisfies 
 more conditions than any atom hitherto imagined. In the 
 first place, it is quantitatively permanent, as regards its 
 volume and its strength, — ■ two independent quantities. It 
 is also quantitatively permanent as regards its degree of 
 implication, whether 'knottedness' on itself or 'linkedness' 
 with other vortex rings. At the same time, it is capable 
 of infinite changes of form, and may execute vibrations 
 
Chap. IV.] TUE VOllTEX-ATOM THEORY. 221 
 
 of different periods, as we know that molecules do. And 
 the number of the essentially different implications of vor- 
 tex rings may be very great without supposing the degree 
 of implication of any of them very high. 
 
 " But the greatest claim to recognition of this theory, 
 from a philosophical point of view, is that its success in 
 explaining phenomena does not depend on the ingenuity 
 with which its disciples ' save appearances ' by introducing 
 first one hypothetical force and then another. When the 
 vortex atom is once set in motion, all its properties are 
 absolutely fixed and determined by the laws of motion 
 of the primitive fluid, which are fully expressed in the 
 fundamental equations. . . . 
 
 [This] "primitive fluid has no other property than 
 inertia, invariable density, and perfect mobility, and the 
 method by which the motion of this fluid is to be traced 
 is pure mathematical analysis. The difficulties of this 
 method are enormous, but the glory of surmounting them 
 would be unique. 
 
 " There seems to be little doubt that any encounter be- 
 tween two vortex atoms would be in its general character 
 similar to those which we have already described. Indeed, 
 the encounter between two smoke rings in air gives a very 
 lively illustration of the elasticity of the vortex rings. 
 
 " But one of the first, if not the very first, desideratum 
 in a complete theory of matter is to explain — first, mass, 
 and second, gravitation. To explain mass may seem an 
 absurd achievement. We generally suppose that it is of 
 the essence of matter to be the receptacle of momentum and 
 energy, and even Thomson, in his definition of his primitive 
 fluid, attributes to it the possession of mass. But accord- 
 ing to Thomson, though the primitive fluid is the only true 
 matter, yet that which we call matter is not the primitive 
 fluid itself, but a mode of motion of that primitive fluid. 
 It is the mode of motion which constitutes the vortex ring, 
 
222 SPECULATIONS AND THEORIES. [Pakt II. 
 
 and wliicli furnishes us with examples of that permanence 
 and continuity of existence which we are accustomed to 
 attribute to matter itself. The primitive fluid, the only 
 true matter, entirely eludes our perceptions when it is not 
 endued with the mode of motion which converts certain 
 portions of it into vortex rings, and thus renders it molec- 
 ular. 
 
 "In Thomson's theory, therefore, the mass of bodies 
 requires explanation. We have to explain the inertia of 
 what is only a mode of motion, and inertia is a property 
 of matter, not of modes of motion. It is true that the 
 vortex ring at any given instant has a definite momentum 
 and a definite energy, but to show that bodies built up of 
 vortex rings would have such momentum and energy as we 
 know them to have is, in the present state of the theory, 
 a very difficult task.'' 
 
 In view of Professor J. J. Thomson's application of the 
 hypothesis in the domain of chemical theory, one might be 
 disposed to emphasize much more strongly the favorable 
 part of this comment ; for no other hypothesis has even 
 attempted an explanation of chemical phenomena as a 
 result of mere motion of matter, while Thomson's explana- 
 tions are strikingly successful. 
 
 Maxwell's objection that the theory does not explain 
 gravitation and mass may easily fail to command equally 
 ready assent with his otherwise favorable view of the 
 theory as a whole. The application of the theory to gravi- 
 tation has been already briefly outlined ; and whatever may 
 be its shortcomings, it is clearly far in advance of all prede- 
 cessors, and has not encountered disproof. 
 
 As to Maxwell's remark with regard to mass, it is not 
 easy to entirely apprehend his position. He clearly inter- 
 prets the term ineiiia as the equivalent of mass, and em- 
 ploys the latter in the sense in which Tiinergety has been 
 
Chap. IV.] THB VOUTEX-ATOM THEORY. 223 
 
 defined in these pages. Consistently with this he also 
 specifically admits that the vortex atom in translatory 
 motion possesses kinetic energy. He says : " It is true 
 that the vortex ring at any given instant has a definite 
 momentum and a definite energy, but to show that bodies 
 built up of vortex rings would have such momentum and 
 energy as we know them to have is, in the present state of 
 the theory, a very difficult task." This comment can be 
 understood only on the supposition that Maxwell regarded 
 mass (kinergety) as an absolute quantity in the sense de- 
 scribed in a later paragraph (cf. Part II. Chap. V.). If, 
 however, as there pointed otit, and as seems probable, energy 
 is merely the power of one portion of matter to move 
 another portion because in motion relatively to it, then 
 kinergety is a relative quantity only. Eemembering then 
 that the " fiuid " exerts no resistance to the motion of the 
 rings, and that it is of uniform density in all rings, it is 
 clear that the theory provides all the explanation of 
 kinergety that any theory can be called upon to give. If 
 the kinetic energy of a body is simply its power of impart- 
 ing kinetic energy to a given body as compared with that 
 of some other body, then we have only to account for the 
 same relative power in the vortex atoms; and this is suffi- 
 ciently done by assigning to them proper relative volumes 
 and by assigning to the "fluid" either kinergety or a 
 property from which this may arise. If the theory postu- 
 lates only one property of matter it can hardly be called 
 upon to explain that property. 
 
 But what is meant by the phrase " as we know them to 
 have " ? We form ideas of the mass or kinetic energy of a 
 body in only three ways : by the acceleration of other 
 bodies, by motion under or against forces, by the sensations 
 accompanying acceleration or retardation of the motion of 
 the object by our muscular action. The second of these 
 gives us no information more than the first except in so far 
 
224 SPECULATIONS AND THEORIES. [Pakt II. 
 
 as we associate with the idea of force the ideas derived 
 from the sensations produced by force. But these are iden- 
 tical in kind with the sensations experienced in accelerating 
 or retarding the motion of freely moving objects. Thus the 
 three methods of forming impressions of the mass or kinetic 
 energy of a body reduce to two. But the sensations in ac- 
 celerating or retarding the free motion of objects (as also of 
 exerting or resisting force) are merely our cognizance of the 
 attendant displacement of portions of our muscular or ner- 
 vous systems. And so far as the material action involved 
 is concerned, this is merely the displacement of one aggre- 
 gation of vortex atoms by another, which is all that we can 
 say of the direct transfer of kinetic energy from one body 
 to another. The sensations produced by light, heat, electri- 
 cal currents, give no means of estimating mass or kinetic 
 energy. In brief then we have no quantitative knowledge 
 of " mass " (kinergety) or kinetic energy save through the 
 power of displacing other bodies. ^Ye have therefore no 
 knowledge of either as an absolute quantity. The theory 
 has then to account only for the relative kinergeties of 
 bodies ; and therefore as the kinergety of the body is only 
 the sum of the kinergeties of its atoms, to account for the 
 relative kinergeties of the atoms. But the latter it does, as • 
 just stated. No absolute density need therefore be ascribed 
 to the fluid. If there be such an intrinsic property as 
 absolute density, it is no part of any physical theory to 
 explain it. "We here reach the confines of the domain of 
 physics. It was not in this, however, that Maxwell criticised 
 the theory, but only in its inability to account for the 
 kinergety of the molecular aggregates called bodies, — a 
 criticism which is untenable if the foregoing remarks are 
 correct. 
 
 In this connection, a possible modification in the funda- 
 mental hypothesis of the vortex-atom theory may be noted. 
 If as suggested at a later point (cf. Part II. Chap. V.), 
 
Chap. IV.] THE VORTEX-ATOM THEORY. 225 
 
 kinergety can be adequately accounted for by the permanent 
 and independent occupancy of space by each minutest por- 
 tion of matter, then this simpler concept may be substituted 
 for kinergety (or inertia or mass) as the postulated property 
 of the perfect fluid. The theory then undertakes the grand 
 task of developing analogues of all the properties of sub- 
 stances, and all forces and forms of energy from modes of 
 motion of vortex rings in a hypothetical matter whose one 
 assumed property is the permanent, continuous, and uni- 
 form occupancy of space, — a task which would appear 
 hopeless and visionary had it not already been well begun. 
 
 Finally, as to its present status we may say that the 
 theory has not yet been successfully assailed. This 
 could be effected only by showing either that its won- 
 derfully simple basal hypothesis is false, or that the 
 mathematical deductions therefrom are defective oi? contra- 
 dictory to fact. Neither has been done, nor has anything 
 been proffered as a substitute for it. The theory has not 
 yet, it is true, been found capable of satisfactorily accounting 
 for several important classes of phenomena, for instance, 
 the electro-magnetic, crystallic, and others ; but this consti- 
 tutes no valid disproof. One might thereby be led to de- 
 spair of its ultimate success, especially in face of the serious 
 mathematical difficulties in the way of its development ; or 
 one might abandon hope of a further outcome from it. But 
 the value of the theory must be judged by what it has ac- 
 complished, not by what we have not yet succeeded in doing 
 with it. And when thus tested, the theory still remains 
 preeminent. 
 
 Of much interest are the following statements of the 
 present attitude taken towards this theory by its two great 
 exponents. The quotations are by permission from per- 
 sonal letters, in response to notes of inquiry which contained 
 statements of views substantially as expressed in the pre- 
 ceding paragraphs. 
 
226 SPECULATION S AND THEORIES. [Part II. 
 
 "London, 18th May, 1898. 
 " Deak Sir, I am afraid it is not possible to explain all the prop- 
 erties of matter by the Vortex-atom Theory alone, that is to say, 
 merely by motion of an incompressible fluid ; and I have not found 
 it helpful in respect to crystalline configurations, or electrical, 
 chemical, or gravitational forces. 
 
 ' ' I am afraid this is the only answer I can give to your letter of the 
 6th, which I have received to-day. I wish I could say a great deal 
 more on a subject which has never ceased to interest me. We may 
 expect that the time will come when we shall understand the nature 
 of an atom. With great regret I abandon the idea that a mere con- 
 figuration of motion suffices. 
 
 "Yours faithfully, 
 (Signed) "Kelvin. 
 
 "Professor S. W. Holm an, 
 "Brookline, Mass., U. S. A." 
 
 "6 ScROPE Terrace, Cambridge, 
 "June 20, 1898. 
 "Dear Sir, . . . With reference to the Vortex- atom Theory, I do 
 not know of any phenomenon which is manifestly incapable of being 
 explained by it ; and personally I generally endeavor (often without 
 success) to picture to myself some kind of vortex-ring mechanism to 
 account for the phenomenon with which I am dealing. In lectures 
 and papers, however, I generally content myself with an illustration 
 which, though it has no claim to the fundamental character of one 
 based on vortex motion, is easily conceived and handled by the mind, 
 and so is more adapted as a guide to research. 
 
 "I regard, however, the vortex-atom explanation as the goal at 
 which to aim, though I am afraid we know enough about the proper- 
 ties of molecules to feel sure that the distribution of vortex motion 
 concerned is very complex. 
 
 " Believe me 
 
 " Yours very truly, 
 (Signed) "J.J.Thomson. 
 
 "Professor S. W. Holman." 
 
CHAPTEE V. 
 NATURE OF ENERGY AND MATTER. 
 
 Is all Energy Kinetic ? — In kinetic energy we recognize 
 two and only two factors or elements ; namely, kinergety 
 and velocity. The kinergety seems to be, or to arise from, 
 an intrinsic property of the body or substance. 
 
 The only respect in which a body undergoes change as it 
 gives up or receives kinetic energy, is in its velocity. The 
 kinergety is unchanged. 
 
 If then the kinetic energy of a body A be expended with- 
 out loss, in imparting kinetic energy to a body B, the only 
 discoverable physid&l change in A or B or surroundings is 
 the change in relative velocity of the two bodies. More- 
 over the only necessary antecedent condition is relative 
 motion of the bodies. 
 
 But if a body A expends its kinetic energy in the pro- 
 duction of heat in the body B, the change in A is absolutely 
 in no wise different from that undergone when the result 
 was kinetic energy in B ; namely, a change in velocity. Or 
 again, if the motion of A be directed upward, so that the 
 kinetic energy is now transformed into gravitation energy, 
 the change in A is still the same as before. The same is 
 true for all other possible transformations of kinetic energy. 
 Likewise in the transformation of any other form of energy 
 into kinetic, the only change in the body in which the 
 kinetic energy is produced is the change in velocity. 
 
 Now there seem to be three possible views of the char- 
 acter of the process called transformation. Eirst, as in the 
 transference of kinetic energy from one body to another 
 
 227 
 
228 SPECULATIONS AND THEOBIES. [Part II. 
 
 the only change in the receiving body is change of trans- 
 latory velocity, so in the case of transformation the result- 
 ing change may likewise be only change in velocity, — 
 although we may be unable to locate the portions of sub- 
 stance or matter undergoing this change of motion. Thus 
 in the transformation of kinetic energy into heat, the heat 
 energy produced may be due merely to increased velocity 
 of random vibration of particles (molecules) of the heated 
 body, — thus remaining kinetic in form, but molecular in- 
 stead of molar. On account of the exceeding minuteness 
 of the molecules, if they exist, we have been unable to ob- 
 tain ocular or other direct demonstration of this hypothetical 
 heat vibration ; so that the kinetic or mechanical theory of 
 heat is based entirely on speculation. Similarly, in the 
 case above cited of transformation into gravitation energy, 
 it is supposable, but not yet demonstrable, that the gravita- 
 tion energy may consist in kinetic energy of a system of 
 impinging particles which form no part of the substance of 
 the body acted upon. Kinetic theories have been framed 
 with more or less completeness for all forms of energy. 
 The allusion to them here is not to raise the question of 
 their acceptability, but to illustrate the degree of attention 
 accorded to the view that all energy may be kinetic. 
 
 Second, we may suppose that some forms of energy are 
 not kinetic in -their nature, but are due to some condition 
 other than motion, imposed upon the substance or matter of 
 the body or its surroundings. The energy would then be 
 due jointly to this condition and to kinergety, as the kinetic 
 form is due to motion and kinergety. There appears to 
 have been no suggestion of any such condition to account 
 for any particular form; and the whole suggestion seems 
 gratuitous and uncalled for, although it can hardly be 
 declared impossible. 
 
 Third, we may suppose that energy is not kinetic at all ; 
 that is, that it is not due to motion, but is something which 
 
Chap. V.] NATURE OF ENERGY AND MATTER. 229 
 
 is transferred from one body or place to another, and mani- 
 fests itself by diverse phenomena, or in other words, assumes 
 different forms, according to certain (unknown) conditions. 
 In this view, transformation must be held to consist merely 
 in so changing the conditions attending the bodies or the 
 energy concerned that the energy shall manifest itself in 
 another manner. The necessary procedure is known for a 
 great variety of cases. But why it is that differences in 
 manifestations should occur is wholly unknown and even 
 unsurmised. 
 
 A conclusive selection between these three views unfortu- 
 nately demands further knowledge or insight than has yet 
 been brought to bear. The second view may be rejected, for 
 reasons stated. Between the first and third, or as we may 
 call them the kinetic and entity views, we may find ground 
 for preference, but hardly for final decision. The weight 
 of argument and the preponderance of sentiment appear to 
 favor the kinetic view. From the foregoing presentation 
 of the kinetic and entity views, a crucial test between them 
 may be seen to lie in ascertaining whether motion alone is 
 transferred when kinetic energy is changed into other forms. 
 If so, then the kinetic view is established, and the principle 
 of conservation of energy becomes merely a numerical rela- 
 tion. If something more than motion is transferred, then 
 the kinetic view would fall, although the entity view would 
 still require further demonstration. In kinetic energy 
 itself, and in the processes of its transference and transfor- 
 mation, there seems to be absolutely nothing to suggest any 
 change except in state or mode of motion, — nothing but 
 changes of velocity. Tor evidence of anything more than 
 this as the essential and complete fact in energy trans- 
 formation, we must then turn to other forms of energy. 
 But of the nature of these other forms we know absolutely 
 nothing. They can therefore furnish no evidence either for 
 or against; and sensation is equally at fault (p. 224). In 
 
230 SPECULATIONS AND THEORIES. [Pakt II. 
 
 corroboration of these remarks it may be noted that all 
 hypotheses of the ultimate nature of energy are kinetic. 
 This, however, is not independent evidence, as it arises from 
 the same causes as the other. There are two considerations 
 which are perhaps contribative to the entity view. One of 
 these is the entirely justifiable habit of regarding energy as 
 a power transferable from one body to another. The other 
 is the principle of conservation. These two ever-conspicu- 
 ous ideas conjoined, easily produce a more or less vivid 
 impression of energy as an entity, that is, as being an irre- 
 solvable thing transferable bodily, so to speak, from one 
 object to another. This idea may be supported by a conserva- 
 tive view of radiation. " Radiant energy " is transmitted 
 through portions of space apparently devoid of any of the 
 elements, that is, of the elementary substances which we 
 recognize through the possession of many different prop- 
 erties by each. Here therefore, it might be asserted, we 
 are free to assume that energy exists apart from substance, 
 or at least that such a position is more logical and conserva- 
 tive than to assume an otherwise unrecognizable substance, 
 the ether, as a medium for transmission. We can but 
 admit, however, that neither of the three considerations 
 advanced affords any real support to the entity view. It is 
 at least as natural to ascribe the " powers " to motion im- 
 posed on matter or substance as to regard it as an irresolvable 
 entity ; and as rational to regard the principle of conserva- 
 tion as a mere numerical law, as to assume that it implies 
 an underlying entity. And as to the assumption of the 
 ether as a transmission medium for radiant energy, this 
 assumption is no whit more gratuitous than our unques- 
 tioned assumption of the existence of any one of our seventy 
 elements. The only essential difference in the case lies in 
 the fact that we find it necessary to ascribe to the ether 
 fewer properties and these in a different degree than to the 
 elements. Matter, as we have repeatedly insisted, is not 
 
Chap. V.] MATURE OF ENERGY AND MATTER. 231 
 
 a thing recognized by the senses but only inferred or rather 
 assumed as the vehicle of energy (cf. pp. 135, 159), and 
 there is just the same kind of reason, for inventing the 
 ether as for inventing any one of the elements. Deferring 
 further detail as to the nature of the ether, enough has been 
 given to show that there is here no definite support for 
 the entity vievi^ of energy. 
 
 In more technical terms the argument is briefly this. 
 In the change of a quantity of kinetic energy into some 
 other form we have a quantity whose " dimensions " are 
 ML^T"^ equated to another quantity (say of gravitation 
 energy), which must therefore have the dimensions also 
 ML^T"^, as we can equate only quantities of the same 
 dimensions. But in the transformation, the mass of the 
 object giving up its kinetic energy is unchanged, and is in 
 general unequal to the mass of the receiving body. The 
 body which gives up its kinetic energy retains its mass or 
 kinergety unchanged. Its change in condition is then only 
 with respect to that which corresponds to the dimension 
 L^T"^, — in this case, velocity squared. But if the change 
 in the acting body is only in respect to L^T"^ whereas the 
 quantity conserved, that is, the energy, is of the dimensions 
 ML^T"^; then, first, the energy must have been transferred 
 to something which has the dimension M, that is, has kin- 
 ergety and is therefore substance, or at least matter ; 
 second, the receiving substance will undergo a change in 
 only such a condition or conditions as correspond to the 
 dimensions L^T~^. If these inferences are correct, then we 
 may conclude that energy cannot exist apart from that 
 which has kinergety, and therefore apart from substance or 
 at least matter, as we have defined these concepts. Further, 
 if the dimension L^T^^ can correspond to nothing but velocity 
 squared, then all energy is kinetic. There is no apparent 
 escape from the first of these conclusions ; is there from 
 the second ? 
 
232 SPECULATIONS AND THEORIES. [Pakt II. 
 
 There is then little to oppose to the view that all energy 
 may be kinetic, but much to uphold it. At best, however, 
 it can be regarded, only as a working hypothesis. As 
 such, it receives most effective support from the vortex-atom 
 theory. The extreme simplicity of the kinetic view renders 
 it of great service to certain minds by supplying concrete 
 analogies to phenomena not otherwise clearly grasped. But 
 it is ever needful to guard against the tendency to gradually 
 install a pleasing analogy as a fixed fact. In the case of 
 the hypothesis in hand, it must be remembered that we 
 have no proof that any one form of energy other than that 
 of molar motion is kinetic. Even for heat, so commonly 
 classed unconditionally as molecular kinetic energy, no 
 specific proof exists other than the general argument just 
 given for all forms of energy. The experiments of Rum- 
 ford and of Davy, for instance, however confirmed and 
 generalized, while disproving the material nature of heat, 
 do not prove that heat is due to a mode of motion. Irre- 
 sistibly as they may be thought to indicate this view, the 
 indication is no stronger or more direct than in any case of 
 transformation of kinetic into any other form of energy. 
 We may be permitted the reiteratipn that the sole, funda- 
 mental consideration of much weight is, that in the trans- 
 formation of kinetic energy, the only discoverable change 
 in the body expending or receiving that kinetic energy is 
 change in velocity. Therefore, to assume that anything 
 other than motion is transferred is gratuitous. But a 
 quantity of molar kinetic energy is convertible directly or 
 indirectly into an equal quantity of any other form, and 
 vice versa. Hence the other forms must also be kinetic, or 
 we must accept as the other horn of the dilemma the un- 
 satisfactory if not untenable supposition that mere motion 
 is convertible into something else or perhaps into several 
 other things of whose nature there is not the slightest 
 trace. 
 
Chap. V.] NATURE OF ENERGY AND MATTER. 233 
 
 Still more concisely, the argument may be stated thus : 
 When the kinetic energy of a body A is transferred, with 
 change of form, to some other object or portion of space, 
 the mass of A remains unchanged. The factor LyT'' is the 
 only one, therefore, which undergoes change in either A or 
 B. But the quantity between which equality exists during 
 the operation is of the dimension MLyT^. Hence, first, the 
 object or portion of space B must possess the dimension M, 
 and therefore must be a substance, not a mere portion of 
 space ; in other words, energy cannot exist apart from mat- 
 ter. Second, unless a quantity which has the dimension 
 L-/T^, and is the square of a velocity, can be changed into 
 one which is not a velocity squared but has still the same 
 dimensions, then all energy is kinetic. 
 
 The kinetic view of energy is not to be regarded as an- 
 tagonistic to the practice now current, and apparently with 
 good results, among certain German writers, of separating 
 energy into factors (called capacity and intensity factors) 
 which differ for different forms of energj'. If energy is 
 kinetic, the mode of motion from which each form arises is 
 characteristic of that form, and the motion, in some forms, 
 is probably not of the substance on which the manifestation 
 of energy appears, but on some external substance. It is 
 therefore to be anticipated that the intensity factors, even 
 if they should prove ultimately to be all of the dimension 
 LyT^, would correspond to very different physical mani- 
 festations. For instance, suppose that heat energy were 
 kinetic, then its intensity factor, temperature, should have 
 the dimension LyT^. Still, as we are able to perceive, not the 
 individual molecular motions or their individual effects, but 
 only the resultants of an immense number of these, it is 
 clear that we should not expect our sense impressions or 
 the physical phenomena of heat to be identical with those 
 of the molar kinetic energy of isolated masses. The case of 
 heat can differ only in degree from that of any other form 
 
234 SPECULATIONS AND THEOBIES. [Part II. 
 
 of energy, but in this one, our imaginations have been 
 supplied with almost the sharp-sightedness of Maxwell's 
 "demons," if not with their delicate alertness, by the 
 kinetic theory of gases. It is to be remembered that the 
 dimensions MLyT^ may be separated into any number of 
 different pairs of factors. Also it is not impossible that 
 some of these pairs may represent actual physical quantities. 
 Hence we may suppose, as the facts prove, that different forms 
 of energy will be recognized through different factors, and 
 that the same form may be recognized and measured also 
 through different pairs of factors in some cases. The classi- 
 fication of the forms of energy in the three groups, space, 
 surface, and volume energies, however useful, seems to the 
 writer to be an outcome of this possible factoring rather 
 than of intrinsic differences in the nature of the energy 
 forms. And it is not unimportant to remark in this con- 
 nection that the postulation of discontinuity, at the point of 
 transformation of energy of one of these classes into that of 
 another, involves the overlooking of an intermediate double 
 transformation ; namely, into and out of elastic energy. 
 The further discussion of these points is, however, beyond 
 the scope assigned to this book. 
 
 But whatever strength we may assign to the kinetic view 
 of energy, \vc can hardly deny that the consensus of opinion 
 of physicists is opposed to its introduction in the presentBr 
 tion of elementary physics beyond the position of an adjunct 
 hypothesis. Confined to this function, it is certainly most 
 useful, but as the substructure of physical teaching its em- 
 ployment is not only unwarrantable but would instil a most 
 flagrant disregard of logic and of the scientific method. 
 
 Objection may be raised that this view of energy is incon- 
 sistent with the concept of matter as purely a carrier of 
 energy, and of substance as endued with energy. There is, 
 however, nothing in the definition to justify any such criti- 
 cism. Matter or substance possessing energy in virtue of 
 
Chap. V.] NATURE OF ENERGY AND MATTER. 235 
 
 an impressed mode of motion is as literally endued with, 
 or a carrier of, energy as if the energy were a pack of iron. 
 
 On Chemical Energy. — On the nature of this form of 
 energy, speculation has made less progress than for other 
 forms, — possibly because the phenomena are complicated 
 by the selective property of atomicity or quantivalence. 
 But aided in our outlook by the vortex theory, we may find 
 it easy and rational to dissociate, or at least to differentiate, 
 atomicity from chemical energy. Atomicity may be wholly 
 independent of chemical energy. It may be due to a purely 
 structural characteristic of the atom in virtue of which two 
 or more atoms can or cannot remain in stable association 
 under the conditions which obtain in substances. Among 
 these conditions we recognize but two as being essential fac- 
 tors in chemical action; namely, molecular impacts, and 
 some force under which the atoms are held in combina- 
 tion, and under or against which chemical work is done. 
 This we will call chemical force and the energy producing it 
 chemical energy; what is their nature ? 
 
 The function of chemical energy as thus differentiated is 
 to maintain chemical force; to do the work attending chemi- 
 cal combination, thus providing the energy which appears 
 as heat or in other ways in chemical processes ; to account 
 for the energy expended in chemical decompositions, etc. 
 With atomicity referred to another cause than chemical 
 energy, the latter manifests a selective power only through 
 the differences in heat of combination of the elements. But 
 this, as will be shown, might arise under a form of energy 
 so universal and simple as gravitation in its law of action, 
 through imposing the condition that the atoms should 
 always approach each' other through a fixed distance. Let 
 us then proceed to develop the clue thus obtained. 
 
 Is chemical energy merely the manifestation of gravita- 
 tion energy when the atomic distances are less than about 
 the atomic diameters ? The fundamental fact is this. At 
 
236 SPECULATIONS AND THEORIES. [Part II. 
 
 distances large relatively to the atomic diameters, tlie atomic 
 force, that is, the only perceptible force betv/'een any two 
 atoms, is that known as gravitation, which follows Newton's 
 law. At distances Ij'ing presiiniably between a very few 
 diameters and a single diameter, the atomic or molecular 
 force is called cohesion, using this term to include also ad- 
 hesion. Of the relation of this force to mass and distance, 
 almost nothing is known, owing to the exceeding minute- 
 ness of the portions of substance and of the distances with 
 which the law deals. There appears to be nothing in the 
 phenomena of cohesion, however, inconsistent with ascrib- 
 ing the force to the same energy which produces the force 
 of gravitation. The distance law of cohesion is perhaps 
 different from that of gravitation, but there is not the slight- 
 est evidence that the distance law of gravitation holds for 
 the small distances at which alone cohesion is manifest. A 
 concrete illustration of the progressive merging of one dis- 
 tance law into another is derivable, obviously enough, from 
 the vortex-atom theory. Lastly, the distances at which the 
 chemical force is manifested are almost certainly smaller 
 than those at which the force is recognized as cohesion. 
 There is no evidence that at these smallest distances, say 
 less than a separation of a single diameter between centers, 
 there are two distinct forces, chemical and cohesive, still 
 less that there are three ; namely, these two with gravita- 
 tion. There seems to be no sufficient reason for not regard- 
 ing the chemical force as merely the cohesion force at shorter 
 atomic distances, with probably a still progressively chang- 
 ing distance law. In face of the radical divergence between 
 chemical and cohesion phenomena this view would wear an 
 air almost of absurdity were it not possible to point to some 
 rational explanation or analogy for atomicity ; but this has 
 been done by Professor Thompson. 
 
 There remains then to be considered the heat of com- 
 bination. Take first the specific case of vapor of sulphur 
 
Chap. V.] NATURE OF ENERGY AND MATTER. 237 
 
 uniting with gaseous oxygen, under constant volume. Sup- 
 pose the sulphur molecules to be monatomic, and uniformly 
 diffused through the combining mass of oxygen. Then the 
 combination would be that of S with O2 to form SOo in the 
 case of each molecule. Under energy acting as we have 
 assumed, every molecule will experience a force towards 
 every other molecule, but owing to the assumed uniformity 
 in distribution, the resultant force on any molecule will be 
 zero. As any molecule of sulphur approaches one of oxygen, 
 their mutual force increases at first, possibly passing through 
 a maximum, but being certainly large relatively to their force 
 of gravitation at greater distances or to the kinergety of the 
 molecules. A relatively large amount of positive work is 
 thus done. A small negative quantity of work is also per- 
 formed, since the average distance of the molecules from 
 their neighbors is increased. The difference between these 
 two quantities is the resultant work done by the chemi- 
 cal energy ; and with the exception of a fraction perhaps 
 left as vibration energy in the atoms, this work must result 
 in an equivalent amount of heat. From the known magni- 
 tude of the cohesion forces and the closer juxtaposition of 
 atoms in chemical combination than when we have to do 
 with cohesion, it is clear that the work done in passing from 
 the cohesion distance to the combination distance must be 
 large. Whether it would be of an order of magnitude to 
 equal the heat of combination can be ascertained only by 
 numerical computation, for which unfortunately we have 
 but crude data. This work must be shown to be not only 
 sufficiently large but also sufficiently constant to correspond 
 to the heat of combustion. As to its constancy, starting 
 with the same initial condition of the combining substances, 
 the average resultant force between a pair of approaching 
 molecules (as S and O2) will always be the same for the 
 same distance between centers ; also the average displace- 
 ment under the force will be the same. Hence the work 
 
238 SPECULATIONS AND THEORIES. [Part II. 
 
 done will be constant, as it should be. Moreover, a differ- 
 ence in initial condition of the substances will make rela- 
 tively small change in the heat of combination, because the 
 resultant force between the combining particles is small 
 throughout the early part of their approach. 
 
 The above-cited simple instance of a gas uniting with 
 another gas to form a gas, all at constant volume, can be 
 easily extended to any combination by taking into account, 
 in the well-known manner, the negative work attending any 
 molecular separations occurring, and the external and other 
 work incident to the operation. 
 
 For a more definite if less general presentation of this 
 hypothesis as to chemical energy, let us use the terms of the 
 vortex-atom theory, adopting also the modified Le Sage bom- 
 bardment theory of gravitation. Then the one form of 
 energy continuously in action causing approach or tendency 
 to approach of all atoms, molecules, and bodies, is the energy 
 of this system of bombarding vortex-corpuscules. This it 
 is which is now suggested as constituting gravitation energy, 
 cohesion energy, and chemical energy, and as exerting what 
 is called chemical force, cohesive force, or gravitation force 
 respectively, according to the degree of proximity of the 
 atoms, molecules, or bodies involved. When in chemical 
 combination, the vortex rings are supposed to be in the 
 closest juxtaposition possible. Thus a molecule of HCl is 
 supposed to consist of a single ring or atom of hydrogen 
 and another of chlorine revolving about one another as 
 described at page 206. The rings have their planes parallel, 
 their diameters sensibly equal, and their distance apart very 
 small relatively to their diameters. In other words, they 
 are practically superposed, except for a separation minute 
 as compared with the diameter of the ring. The Le Sage 
 force here is the chemical force, and is large relatively to 
 the kinergety of the atom ; that is, under it the atom would ac- 
 quire a high velocity in traversing a short distance. In all 
 
Chap. V.] NATURE OF ENERGY AND MATTER. 239 
 
 molecules the atomic distances are thus small. In solids, 
 even at their densest, the distances between centers of 
 molecules presumably average more than the molecular 
 diameter, and therefore more than the atomic diameter also. 
 The Le Sage force between molecules at this distance, and 
 up to somewhat greater ones, is, in general, the force of 
 cohesion. Vortex molecules would be capable of somewhat 
 closer packing than this, but the resulting solid would not 
 be isotropic, and the molecules would have even less free- 
 dom than the small degree apparently pertaining to the 
 molecules of a solid at ordinary temperatures. As the 
 molecular distance increases slightly, the cohesion dimin- 
 ishes, and we have the analogy to the liquid state. And as 
 the distance becomes several diameters, the Le Sage force 
 passes gradually over from cohesion into gravitation, in 
 which it presumably exists in perfect gases. 
 
 The distance law of the Le Sage force between a pair of 
 simple vortex rings ought to be capable of mathematical ex- 
 pression. Starting with the two rings superposed, and draw- 
 ing them apart perpendicularly to their planes, the force 
 might perhaps lessen at first nearly in proportion to the dis- 
 tance between the planes. The rate of diminution would, 
 however, increase with the distance, so that at a few diame- 
 ters the law would approximate to that of gravitation to 
 which it would conform with ever-increasing closeness as the 
 distance grew larger. Su^jpose now that the rings be with- 
 drawn from one another in a different manner. Let them 
 be slid past one another in their own planes. At the start 
 the Le Sage force will be very great, but it will diminish 
 rapidly at first as the rings leave the position of immediate 
 superposition, remaining rather large, however. It will 
 diminish rather slowly as the rings slide over each other 
 always in contact at two points, until the final point of out- 
 side contact is reached. The Le Sage force in this position 
 should tally with or be somewhat greater than the force of 
 
240 SPECULATIONS AND THEORIES. [Pakt II. 
 
 cohesion of solids. This appears probable from the close- 
 ness of the rings at the point of contact, and from the rela- 
 tively small section of a vortex ring compared with its 
 diameter. 
 
 The law of the Le Sage force as a function of the kinerge- 
 ties of the rings, is presumably not entirely simple except, 
 perhaps, for large distances ; for the bombarding corpuscules 
 impinge on the surface only of the rings, and the degree of 
 implication of vortex atoms of different elements may be 
 widely divergent. In the characteristic linkage or be- 
 knottedness of the atoms, the seat of the observed diversity 
 of the cohesion of different substances is probably to be 
 sought, as well as in the structure of the molecule. 
 
 Turning to a specific case, consider a mixture of hydrogen 
 and oxygen gases. Any molecule H, and any molecule Oj 
 will find themselves under sensibly balanced forces during 
 the greater portion of any free path ; but as two such mole- 
 cules approach within a few diameters, the resultant Le Sage 
 force will become sensible and will rapidly increase. The 
 molecules will therefore acquire increasing kinetic energy 
 relatively to each other until, if moving in the right direc- 
 tion, they collide. Prom the impact they will rebound with 
 altered velocities and vibrations. They cannot remain in 
 association because the vortices HjOa do not form a stable 
 system, as shown at page 211 ; although it is conceivable 
 that under very favorable and rare conditions they might 
 for a brief time remain in combination as Hj-O-O (hydrogen 
 dioxide), to be broken up at some shortly ensuing collision. 
 The velocity and corresponding energy acquired by the 
 molecules in approaching must be given up during reces- 
 sion, the energy being restored to the Le Sage system. The 
 vibration energy will be subject to the Boltzmann restitu- 
 tion law (p. 177). Thus, on the average, no heat production 
 or absorption will occur at an impact. This is the case at 
 ordinary temperatures. At higher temperatures the vio- 
 
Chap. V.] NATURE OF ENERGY AND MATTER. 241 
 
 lence of impact will be greater. Beyond a certain point 
 dissociation will occur, as stated at page 214, some of tlie 
 molecules O2 or H2 being separated into their constituent 
 atoms 0,0 and H,H respectively. Some negative work 
 will therein be done, heat being transformed into Le Sage 
 energy. Suppose now an Ho and a dissociated to be 
 approaching one another, as were the Hj and the O2 in the 
 former case. If the direction of motion and the relative 
 position of the vortex rings be unfavorable for combination, 
 the H2 and the will rebound just as did the Hj and 
 the 02- But if the conditions be favorable, the H„ and 
 will remain in rotation about each other permanently ; 
 that is, the chemical combination will take place, and HjO 
 will be formed. During the approach, work will be done 
 under the Le Sage force, here recognized as chemical force, 
 and the translatory and vibratory energy of the system H2O 
 will be correspondingly increased. This translatory energy 
 is heat, and the vibration energy will presently become heat. 
 The Le Sage force between the Hj and the would pre- 
 sumably be less (probably one-half) at equal distances than 
 between the H2 and the O2, as the O2 has twice the mass and 
 volume of the single atom 0. But the closeness of ultimate 
 approach in the combination would undoubtedly be greater in 
 the case of the combination, and the whole work of approach 
 results in heat. The observed heat of combination of a 
 quantity of hydrogen and oxygen would then be the equiva- 
 lent of the algebraic sum of the Le Sage work of dissociation 
 and of combination, together with any other work incidentally 
 performed. The elements of the action have been stated 
 with perhaps unnecessary fulness. Of the application to 
 more complex cases certainly nothing more need be said 
 than that they involve only due consideration of all the 
 work, both positive and negative, involved in the action. 
 
 By the introduction of the Le Sage force as a factor in 
 chemical combination, a new element of stability is im- 
 
242 SPECULATIONS AND THEORIES. [Pakt II. 
 
 parted to the molecule, but at the same time the violence of 
 impact is also increased in a ratio nearly or quite correspond- 
 ing ; the necessity for the selective function of the vortex 
 rings therefore still remains, as well to account for perma- 
 nence as for valency. This extension of the Le Sage theory 
 seems then to supplement the vortex-atom theory of chemical 
 action in just the way needed to give it completion. 
 
 Without the aid of the vortex hypothesis, one could hardly 
 arrive at the view of the nature of chemical action above 
 presented, but this point once reached, we may easily see 
 that the idea of the unity of chemical, cohesion, and gravi- 
 tation energies does not depend on the tenability of that 
 hypothesis. Almost any theory which would account for 
 one of these three forms would presumably account also for 
 the others, provided only that some consistent explanation 
 for valency could be found. A view so comprehensive can 
 but command interest even in so crude a form, and must 
 doubtless have often suggested itself. Its development 
 through the vortex hypothesis, if made, has escaped the 
 author's attention. 
 
 The theory of chemical energy thus suggested can be- 
 come entitled to weight only through mathematical and 
 quantitative confirmation. This the author is unable to 
 attempt beyond the following brief discussion. In view 
 of this fact, the whole suggestion is given space only as 
 a tentative speculation of small weight but of perhaps some 
 interest. 
 
 Assuming the Newtonian law of gravitation to hold for 
 the chemical force at all distances between atoms, we may 
 readily deduce an expression for the work done by the 
 chemical energy in causing the approach of atoms in chemi- 
 cal actions. Similar expressions may be derived for the 
 action under other laws of force, but they need not be 
 entered into here. 
 
 The work w, which will be done by the energy which 
 
Chap. V.] NATUBE OF ENERGY AND MATTER. 243 
 
 causes the approach of a pair of atoms, the distance law of 
 the force being F(s), will be, 
 
 = rF(s)ds. 
 
 If this energy be that of gravitation, and there be N pairs 
 of molecules or radicals, one member of each pair consist- 
 ing of n-i atoms of atomic weightal a^ and the other member 
 having jij atoms of atomic weightal aj, a^ and a^ being 
 expressed in grammes, then the total work of approach W 
 from a distance s = cZ to a distance s = x will be given by 
 
 W = n,a, ■ )(«a, • C!N" 
 
 ■X" 
 
 where C is the " gravitation constant," that is, the attraction 
 between two " masses " of one gramme each at a distance of 
 one centimeter between centers. The distances s, d, and x 
 are measured between the centers of the atoms, and the 
 vortex rings are assumed to have their planes parallel, and 
 the line joining the centers is assumed to be perpendicular 
 to these planes. The formula is not assumed to be uni- 
 versal, but is sufficiently general for the present purpose. 
 In any chemical reaction, the heat evolved, expressed in 
 ergs, will be the sum 2W of all the separate quantities of 
 combination work done, minus the work SW of separating 
 the initial molecules into the radicals which unite to form 
 the final molecules, and minus any external work W" per- 
 formed. That is, HJ = 2W - 2W' - W". The quantity 
 W is of the same kind as W, and would be found from an 
 expression similar to that for W. 
 
 As the distance x, the closeness to which the atoms must 
 approach to permit of the performance of an amount of 
 work consistent with the observed heats of combination in 
 specific cases, is not known, we may apply the above expres- 
 sion to deducing its approximate value, from which we may 
 then proceed to deduce an approximation to the order of 
 
2-t4 SPECULATIONS AND THEORIES. [Pakt II. 
 
 magnitude of the thickness of a vortex ring, as this cannot 
 be greater than x. AVe shall make the assumption, justified 
 by the sequel, that the chemical energy expended at dis- 
 tances greater than D is negligible. 
 
 The value of C is 6.5 • 10"^ c.g.s. The value of E" may be 
 approximated as follows : The nvimber of atoms of hydro- 
 gen in one gramme is 2 N, since the molecule is Hj. The 
 gramme of liquid hydrogen at its boiling-point has a volume 
 of about 14 cc. Assuming then that in liquids the molecules 
 are arranged in square order, with centers at a distance D, 
 or in an equivalent manner, ]SrDy2 = 14, and N = 28/D'. 
 Adopting D = 10-«, this gives N = 2.8 ■ 10^^. As all mole- 
 cules and atoms have roughly the same diameter, we may 
 check this value by other liquids, e.g. water. Here N" mole- 
 cules have the volume, in the liquid state, of 18 cc. ; so that 
 N = IS/D' = 1.8 • 10^. Both numbers are presumably too 
 small. "We will adopt as a value IST = 3 • 10^. 
 
 We will take the case of HjO, formed from the direct 
 combination of gaseous hydrogen and oxygen. As the 
 approximation can at best be but rough, we may neglect 
 W In this case, W is the work of separation of O2 into 
 0,0. This is unknown, but is unlikely to be of a greatly 
 different order of magnitude from W, that is, from the work 
 H2 -|- 0, since the molecular operation is apparently of about 
 the same scale of magnitude. Other considerations, such 
 as the relative magnitudes of the heats of formation of 
 H2O and H2O2, and of CO and COj, seem to indicate that 
 the order of magnitude of HJ, in simple cases like these, 
 does not differ much from that of %^Y'- Hence we may 
 neglect the unknown quantity 2W in the present pro- 
 visional computation. Then for HjO under the distance 
 law s-', we have, HJ = 6.8 • 10^ ; 4.2 ■ 10' = 2.9 ■ lO'^ ; Wi = 2 ; 
 «! = 1 ; ?i2 = 1 ! a2 = 16. Substituting the foregoing values 
 in the above expression for HJ, and solving, we have, 
 X-' - D-^ = 4 • 10«. As D = 10-^ D-i is clearly negligible 
 
Chap. V.] NATURE OF ENERGY AND MATTER. 245 
 
 compared with a;-'. Hence, approximately, x = 2.5 ■ lO-"* cm. 
 Computations from HCl, CO, CO2, and other compounds 
 give exponents differing by only one or two units from 
 this. The neglect of W can hardly be expected to intro- 
 duce a greater error than a few units in this exponent. 
 Hence we may suppose 10"" to be about the order of 
 magnitude of x, so that the thickness of a vortex ring is 
 presumably not greater than 10"" cm. Or, as D = 10"*, 
 x/B = 10-^1 If the force increases with the distance at 
 a less rapid rate than under the Newtonian law, then the 
 exponent will be much greater than 44, and the ring corre- 
 spondingly thinner. Interesting lines of inquiry into the 
 relative closeness of approach of atoms in different com- 
 pounds, as to the heats of energy of combination of H -f H, 
 etc., and as to the distance law itself, naturally here sug- 
 gest themselves. 
 
 The foregoing result at first inclines one to utter incredu- 
 lity. A spider-web ring with a diameter equal to that of 
 the earth's orbit would be of more substantial proportions. 
 We were prepared, it is true, by one of the early deductions 
 of the vortex-atom theory of chemical action to find the 
 ring-thickness small as compared with the diameter, but 
 we had hardly, stretched the imagination to this figure. 
 Still, only a moment's consideration is needed to show that 
 this is precisely the condition absolutely demanded by the 
 Le Sage theory of gravitation. To possess the degree of 
 permeability demanded by the Le Sage theory, so that the 
 force of gravitation, shall not be modified by intervening 
 portions of substance, or, in other words, so that Newton's 
 law of masses shall hold universally within observed limits, 
 the rings must be enormously thin relatively to their diam- 
 eter. Let us attempt a rough numerical estimate of the 
 lower limit of this permeability. We are probably justified 
 in assuming that the gravitation between bodies on oppo- 
 site sides of the sun is interfered with by the screening 
 
246 SPECULATIONS AND THEORIES. [Part II. 
 
 action of the sun by less than one part in 10^. Then, sup- 
 posing the average distance between the centers of atoms 
 to be one diameter, the shadow of a row of particles as long 
 as the sun's diameter must be less than 10~^ of the area 
 enclosed by a single ring. Assuming the rings most favor- 
 ably arranged, this shadow could not exceed the sum of the 
 shadows of so many separate rings. The row would con- 
 sist of 1.5 • 10" H- 10-8 ^ 15 . 10". The shadow of this 
 row of atoms must not exceed 10"° of the area enclosed by 
 a single ring. The width of the ring must then be less 
 than 10"^ of the diameter. This rough estimate is almost 
 certainly very much too large, but so far as it has any 
 weight it sustains the deduction above made. 
 
 It is quite supposable that the conditions of stability of 
 chemical compounds demand the same degree of thinness, 
 but there seem to be no available means of applying that or 
 any other test, and the discussion must rest at this point. 
 
 On the Ether. — The theory of the ether has received such 
 careful and repeated exposition in connection with the 
 phenomena of light and of electromagnetic radiation in 
 general that a restatement here would be superfluous. A 
 word may be said, however, as to the classification of ether 
 relatively to substance and matter. 
 
 Stress has been laid in the foregoing pages on the point 
 that substance is that which we assume, or at best infer, as 
 existing in space, because we find there such quantities and 
 manifestations of energy as indicate definite specific capaci- 
 ties for energy. Such capacities, differing permanently in 
 degree, we are not able to reconcile with our concept of 
 pure space ; hence we impose on space, substance, to serve 
 as the seat of these capacities and as the carrier of energy. 
 It must be remembered that the capacities themselves are 
 not directly observed, but are only inferred from the 
 energy present and from its effects on other portions of 
 substance or on our own senses. With these points in 
 
CiiAF. v.] NATUIiE OF ENEliGY AND MATTER. 247 
 
 mind, does it appear that the concept of the ether is in 
 any way different in kind from that of substance? The 
 answer to this query appears to be in the negative, so that 
 the ether must be classed as either substance or matter. 
 The ether is primarily that which is inferred as exist- 
 ing in space to serve as the carrier of radiant energy. 
 Eadiant energy is transmitted through it at a rate deter- 
 mined by some property of the ether and not by the 
 properties and conditions of the radiating body solely. 
 This fact of a constant velocity of transmission, — the same 
 for radiations of the most diverse character and intensity, • — • 
 supplies the same warrant in kind for adopting the assump- 
 tion of the existence of ether that we have for adopting that 
 of any of the elements. The difference, fundamentally and 
 apart from hypothesis as to mode of propagation, lies solely 
 in the small number of energy forms which we are obliged 
 to find capacities for in the ether. But the apparent 
 absence of gravitation, of chemical action, and so on, is no 
 more a reason for the exclusion of the ether from the list of 
 substances than obtains between different recognized sub- 
 stances through lack of magnetic capacity, and other differ- 
 ences. And if we turn to the theories of energy propagation 
 in the ether, we find that they impose ujpon the ether the fun- 
 damental properties of substance ; namely, kinergety and 
 elasticity. From the point of view adopted in this book, then, 
 there is no sufficient reason for not assigning to the ether a 
 place as substance. Its precise relation to the elements need 
 give us no concern until it is found taking part in chemical 
 action. Whether its elasticity arises after the manner of 
 that of substances, or in some other way, is a question 
 whose solution seems almost hopelessly remote. It seems 
 however to be admitted that a grained structure of the ether 
 might not be inconsistent with its observed function, pro- 
 vided that the granulation were of a sufficient order of fine- 
 ness. Be these things as they may, however, there appears 
 
248 SPECULATIONS AND THEORIES. [Part II. 
 
 to be no reason for the traditional isolation of the ether as a 
 concept quite apart from that of substance, and utterly 
 foreign to it, — as a something vague and mystical, eventu- 
 ally to step forth from fairyland and furnish the explanation 
 of all the properties of " gross " substances. 
 
 On the Nature of Force. — The mode of action by which 
 energy exerts force varies presumably with the form of 
 energy exerting the force. As we know nothing of the 
 nature of these sundry forms save by hypothesis, so our 
 knowledge of their actions, called forces, can be only of a 
 hypothetical character. The views which have been pre- 
 sented incidentally to the discussions of the forms of enesgy 
 will be briefly summarized here. 
 
 The most elementary of the forces seems to be that of 
 elasticity. We trace this back to the molecules or ultimate 
 particles of substances. There, we have but one rational 
 hypothesis to account for it in terms of something appar- 
 ently more simple. This refers it to the tendency of vortex 
 rings to resume their normal shape when distorted, as 
 through collision. This tendency is due to the energy of 
 the vortex motion ; and the energy which the distorted ring 
 can give out is the translatory energy unequally distributed 
 through it as a result of the impact of other rings with 
 which it is or has been in collision. Further, as the impart- 
 ing of motion from one ring to another can take place only 
 through simple displacement of portions of the inert 
 inelastic "fluid," or matter, by another portion, elastic 
 force reduces ultimately to mere displacement. 
 
 Under the hypothesis herein advanced, gravitation, 
 cohesion, and even chemical force, are supposed to be due 
 to a bombardment process analogous to that so generally 
 accepted as accounting for the pressure of gases. In the 
 process, however, an essential and fundamental element is 
 seen to be the exertion of elastic force by the molecules or 
 particles of the systems involved. It is indeed this which 
 
Chap. V.] NATURJE OF ENERGY AND MATTER. 249 
 
 constitutes the observed force, the bombardment being 
 merely the process by which it is called into play. 
 
 All explanations of electrical, magnetic, and electromag- 
 netic forces, and of all forces involved in radiation, likewise 
 reduce in the end to elastic force ; and the list of forces is 
 thus exhausted. 
 
 Thus we easily arrive at the hypothesis that the force act- 
 ually exerted by all forms of energy is really of one kind ; 
 namely, elastic force. The difference between the various 
 forces, then, consists only in the different modes by which 
 the several forms of energy bring into play this elastic force. 
 A little reflection will show that this view may be arrived 
 at from rather more general grounds than the above; so 
 that it has really more weight than some of the hypotheses 
 incidental to the foregoing deduction. 
 
 If now we revert to elastic force under the aspect above 
 presented, we arrive at the Ultima Thule of speculation in 
 the deduction that all force is in the end mere displacement 
 of the continuous hypothetical matter. Such speculations 
 afford a unity and breadth of view of physical phenomena 
 not otherwise attainable, and in so far are good. But their 
 usefulness depends largely on the rationality of the process 
 by which they are deduced, and on the firmness with 
 which their purely hypothetical character is held in mind. 
 
 On the Nature of Matter and Kinergety. — In continuation of 
 the discussion of Part I., Chapter XI., it may be remarked 
 that if the sole property of substance for which we are 
 to find a counterpart in matter, or which we are to transfer 
 to it is kinergety, then speculation on the nature of matter 
 resolves itself into speculation as to the nature of kinergety. 
 
 When we push against an object such as a table resting 
 firmly on the floor, and the object does not move, we feel 
 a res"istance to the hands. We know that this arises 
 from a resistance to motion manifested by the table, due 
 ultimately to a frictional resistance between the table 
 
250 SPECULATIONS AND THEORIES. [Part II. 
 
 and the floor. Even if the table slides, we feel a like 
 resistance, and know that it is due to the same friction, at 
 least in part. But if the table were mounted on a friction- 
 less level railway, we should experience in accelerating its 
 motion a resistance entirely indistinguishable in kind from 
 that encountered in the other cases. The motion would be 
 different, but the sensation itself would be the same. In 
 starting into motion a cannon ball suspended by a long 
 cord we should experience a like resistance. As there is no 
 external resistance in the last two, the assumption is not 
 unnatural that the object itself exerts a resistance. And 
 this statement is true in so far as it is interpreted to mean 
 only that the elastic energy in the body exerts a resistance 
 against the hands. It is not true, as has been shown (cf. 
 p. 140), of the matter composing the table. ISTeverthe- 
 less it is undoubtedly by supposing matter even when free 
 to be capable of resisting motion, that there is commonly 
 ascribed to matter, quite unconsciously, an attribute which 
 from its vagueness and unreality is difficult to put into 
 words. 
 
 This vague attribute is a property in virtue of which 
 matter is supposed to be able to resist motion or when in 
 motion to resist retardation. By some persons it is associ- 
 ated with the idea of inertia, by others with that of mas.s. 
 It properly belongs with neither of these. If it corresponds 
 to any real physical fact at all, that fact must be related 
 to the property of kinergety. As matter is non-resistant, the 
 above process of demonstrating this property is incorrect, 
 and therefore the attribute may be illusory. On the other 
 hand, " kinergety," and " the capacity for kinetic energy," 
 are but names for a property of substance for which we seek 
 naturally some intelligible explanation in matter. This 
 might lead us back to the same vague attribute as before, 
 which as nearly as it can be expressed, perhaps, is implied 
 by such terms as substantiality, denseness, solidness, and the 
 
Chap. V.] NATURE OF ENERGY AND MATTER. 251 
 
 like. The idea may also possibly be suggested by the phrase 
 absolute density, and if so, that phrase may at the same 
 time serve the added purpose of pointing the inconsistency 
 of the attribute with modern physical science where the 
 relative so completely displaces the absolute. 
 
 Now our sensation of force exerted on us from without 
 is our cognizance of the displacement of some portion of 
 our muscular or nervous system and of the effort exerted to 
 resist that displacement. Thus our sense recognition of 
 kinergety depends on the power of one portion of substance 
 to displace another portion relatively to which it is in 
 motion ; namely, some part of our body. We therefore learn 
 nothing further of the nature of kinergety from sensation 
 than we already know from the action of inert substance. 
 Hence whether the supposed property of "absolute density" 
 inferred in the preceding paragraph is or is not real, we 
 have no ground for it in sensation. 
 
 It appears then that all that is to be accounted for in 
 kinergety is the power of one portion of substance to dis- 
 place another relatively to which it is in motion. A sub- 
 stance of which every minutest portion permanently occupies 
 its own quota of space, would seem to possess this property. 
 Unless however these minutest particles were in every way 
 indistinguishable from one another they would be thereby 
 shown to have some other remaining property. But we have 
 supposed that matter has no other recognizable propertj- 
 than kinergety. Therefore these minutest particles or 
 rather portions must be indistinguishable. Continuity is 
 also demanded in that out of which substances are made up, 
 as otherwise, being devoid of force, its aggregations into 
 particles of substance could not be permanent. Continuous, 
 uniform, and permanent occupancy of space appears then to 
 be the ultimate and sole property demanded of matter in 
 order that it should fulfil all that we know of it. 
 
 The very natural objection to such a view of matter as 
 
252 SPECULATIONS AND THEORIES. [Part II. 
 
 that just presented is, of course, that we are unable to 
 satisfactorily grasp such a concept. But the difficulty is 
 hardly greater than for the abstract time and space concepts 
 or for all fundamental abstract ideas. We may be unable 
 to picture to ourselves mere permanent occupancy of space, 
 but no more are we able to imagine what can constitute 
 absolute density, "mass," "inertia," or whatever other 
 substitute we may be accustomed to think of as the ultimate 
 property of matter; moreover we have here to do with a 
 psychological rather than a physical point. It is in no re- 
 spect more difficult. We must not overlook the fact, however, 
 that the view is merely the result of pushing speculation to 
 the limit, so that it is at best only of tentative character; 
 and also that its acceptance or rejection is not necessary to 
 a competent knowledge of physics. Still, the unity of 
 physical ideas of which this view forms the key-stone is not 
 without its value. Briefly it is as follows : — 
 
 Matter : that which is assumed to uniformly and continu- 
 ously fill infinite space. Atoms : permanent aggregations 
 of matter, differentiated from matter by some mode of 
 motion {e.g. vortex motion), and from each other by 
 structural characteristics (e.g. volume, size, shape, etc.). 
 Molecules : congeries of atoms. Forms of Energy : due to 
 modes of motion of the atoms or molecules themselves, or 
 of special atoms or corpuscules impinging on them. Force : 
 the action of one or more forms of energy, producing ten- 
 dency to motion in the molecules or atoms, or their aggre- 
 gates acted upon. Work : the complete action of energy 
 when this motion takes place. Substance : atoms, molecules, 
 or molecular aggregates. Bodies and Objects : limited por- 
 tions of substance. The Ether: presumably a substance 
 of very special and few properties. Properties : char- 
 acteristic of substances, derived from atomic structure or 
 from the action of energy. 
 
INDEX. 
 
 Acceleration, 14, 15, 16. 
 Action, 18, 51, 163. 
 Actual energy, 126. 
 Adhesion, 4 (see Cohesion). 
 Atom, 137, 155, 193, 252. 
 
 dimensions of, 180, 243, 245. 
 
 Le Sage's supposed form of, 185, 
 216. 
 
 vortex, 202, 205. 
 Atomic mass, 153, 155. 
 Atomic "weight, 153, 155. 
 Atomic weightal, 155. 
 Atomicity, 206, 211, 212, 213, 235, 242. 
 Attraction, 50. 
 
 Balance (see Equal-arm balance and 
 
 Spring balance) . 
 Ball, 199. 
 Beknottedness in vortex rings, 201, 
 
 205. 
 Body, 6, 252. 
 
 BOLTZMANN, 175, 177, 188, 240. 
 Boyle's law, 178, 204. 
 
 Capacities, 8, 9, 55, 135, 138, 140, 146, 
 
 148, 160, 246. 
 Capacity, 7. 
 Cause and effect, 3, 42. 
 C. G. S. System, 107, 119. 
 Chemical action, 4, 32, 70, 193. 
 
 action, time rate of, 215. 
 
 action, vortex theory of, 205. 
 
 affinity, i, 32, 206, 2.35. 
 
 combination, work of, 243 (see 
 Combination, heat of). 
 
 energy, 31, 70, 85, 87, 136, 212, 235. 
 
 energy, speculation on, 235. 
 
 force, 235, 248. 
 
 substance, 9. 
 
 Clausius, 175, 181. 
 Clock-weight, 84, 121. 
 Cohesion, 4, 30, 217, 236, 238, 242. 
 248. 
 
 energy, 30, 136, 236, 238, 242. 
 Collision, 26, 32, 55. 
 Combination, heat of, 70, 235, 242. 
 Connotation, 7. 
 
 Conservation of energy, 67, 71, 73, 
 77, 229. 
 
 of matter, 71, 75, 156. 
 Constant body, 59. 
 Corpuscules, 182, 215. 
 
 Davy, 74, 232. 
 
 Definitions, comments on, 158, 166. 
 
 Denotation, 7, 159, 162. 
 
 Density, 224, 251, 252. 
 
 Descartes, 76. 
 
 Direction, 13. 
 
 Dissipation of energy, 94. 
 
 Dissociation, 206, 214. 
 
 Distance, 12. 
 
 Dyne, 107. 
 
 Eifort, 22 (see Energy). 
 Elastic collision, 26, 32, 42, 55, SO, 172. 
 energy, 30, 43, 83, flO, 93, 94, 122, 
 
 138, 141, 172, 181, 250. 
 force, 248. 
 Elasticity, 4, 30, 61, 156, 200, 218, 247, 
 248. 
 of vortex rings, 200. 
 Electrical energy, 4, 33, 34, 69, 70, 76, 
 
 84, 85, 87, 89, 136, 249. 
 Electricity, 33, 70. 
 Electro-magnetic action, 4, 35, 249. 
 Elements, 7, 140, 213. 
 Energy, 18-24, 53, 59, 67, 73, 77, 92, 
 253 
 
254 
 
 INDEX. 
 
 :26, 135, 140, 143, 164, 223, 248, 
 
 252. 
 chemical (see Chemical energy) . 
 classification of, 37, 38, 234. 
 cohesion (see Cohesion energy) . 
 conservation of, 67, 71, 73, 77, 229. 
 criterion of, 20, 25, 52, 72. 
 definition of, 20, 50, 92, 160. 
 discontinuity of, 234. 
 elastic (see Energy of elasticity) . 
 electrical (see Electrical energy), 
 expended in maintaining motion, 
 
 24 (see Work) . 
 expended in producing accelera- 
 tion, 24 (see Work), 
 factors of, 75, 124, 233. 
 force as an index of, 52. 
 forms of, 2.5-37, 234, 252. 
 heat (see Heat) . 
 Is it all kinetic? 38, 227. 
 kinetic (see Kinetic energy), 
 kinetic view of, 227, 234. 
 magnetic (see Magnetic energy), 
 measurement of, 69, 75, 124. 
 mechanical, 38. 
 muscular, 22. 
 nature of, 21, 37, 127, 136, 169, 170, 
 
 227, 235. 
 of elasticity, 30, 43, 83, 90, 93, 94, 
 
 122, 138, 141, 172, 181, 234, 248. 
 of gravitation (see Gravitation 
 
 energy), 
 of motion, 21 (see Kinetic energy) . 
 of position, 130. 
 of sound vibration, 34. 
 potential, 38. 
 producing force, 40, 43. 
 quantity of, 69, 118. 
 radiant (see Radiant energy), 
 test for, 20, 25, 52, 72. 
 theories of, 227, 235. 
 transference of, 117. 
 transformation of, 67, 78, 90, 117, 
 
 227. 
 Energetic, 59. 
 Energia, 59. 
 Energic, 60. 
 Entity, 42, 229, 2.30. 
 Equal-arm balance, 152. 
 
 Equilibrium, 99, 120. 
 Erg, 107, 119, 125. 
 Ether, the, 36, 230, 246, 252. 
 Extension, 143, 160, 225. 
 
 Fluid, perfect, 193, 201. 
 Foot-pounds, 119. 
 
 Force, 40-54, 67, 70, 76, 92, 90, 100, 
 113-116, 118, 131, 160, 162, 217, 
 248, 252. 
 
 as an index of energy, 52. 
 
 confusion of, with energy, 53, 54, 
 71, 76. 
 
 definition of, 40, 41, 71, 76, 160, 
 163. 
 
 definition of, by Newton, 73. 
 
 direction of, 42, 112. 
 
 exertion of, by energy, 40, 92, 141 . 
 
 measurement of, 76, 96-111. 
 
 nature of, 40-,54, 171, 172, 177, 182- 
 192, 217, 248. 
 
 of a blow, 53. 
 
 of motion, 53. 
 
 quantity of, 96, 165. 
 
 recognition of, 44, 52. 
 
 standards of, 100, 107. 
 
 terms relating to, 54. 
 Forms of energy, 25-37. 
 Free motion, 16, 18. 
 Friction, 45. 
 
 external, of solids, 46. 
 
 internal, of fluids, 49, 177. 
 
 internal, of solids, 47. 
 
 Gases, vortex-atom theory of, 202. 
 Gravitation, 4, 28, 31, 51, 70, 83, 145, 
 
 149, 182, 215, 217, 235, 246. 
 acceleration by, 101. 
 constant, 243, 244. 
 energy, 28, 31, 43, 51, 69, 70, 75, 
 
 82, 83, 90, 92, 136, 146, 148, 
 
 228, 235. 
 energy, nature and location of, 
 
 28, 90, 182. 
 Le Sage theory of, 182, 213, 216, 
 
 238, 245. 
 vortex-atom theory of, 215, 238, 
 
 245. 
 Gravity, 82. 
 
INDEX. 
 
 256 
 
 Hamilton, Sik William, 2. 
 Heat, 4, 29, 32, 35, 69, 70, 74, 76, 83, 
 
 85, 88, 91, 92, 93, 94, 127, 136, 
 
 228, 232. 
 nature of, 29, 228, 232. 
 Helmholtz, 76, 193, 195. 
 Hypothesis, function of, 169. 
 
 Impulse, 14. 
 Inert, 10, 140. 
 
 Inertia, 19, 28, 141, 143, 193, 222, 224, 
 250, 252. 
 
 Joule, 75, 83, 175. 
 
 Kelvin, 175, 180, 188, 193, 195, 201, 
 
 226. 
 Key, 2. 
 
 Kinematics, 12. 
 Kinerg, 60. 
 Kinergia, 60. 
 Kinergic, 60. 
 Kinergy, 60. 
 
 Kinergety, 55, 59, 106, 108, 138, 143, 
 144, 146, 148, 149, 153, 154, 156, 
 193, 202, 222, 223, 227, 247, 249, 
 250. 
 definition of, 59, 60. 
 measurement of, 63, 108, 111. 
 primary measurement of, 62. 
 proportional to mass, 150, 217. 
 proportional to weight, 63, 109, 
 
 150, 216. 
 proportional to weightal, 66. 
 quantity of, 62. 
 standard of, 63. 
 unit of, 63. 
 vs. mass, 61, 153, 250. 
 Kinergeties, graded set of, 62, 111. 
 Kinetic energy, 25, 55, 69, 83, 84, 88, 
 92,93, 114,223, 227. 
 apparatus for demonstrating law 
 
 of, 55. 
 capacity for, 59 (see Kinergety). 
 definition of, 27. 
 formula for, 65. 
 law of, 59, 64. 
 measurement of, 58, 65. 
 molar, molecular, atomic, 27. 
 
 of rotation, 27. 
 
 quantity of, 58. 
 
 reference point for, 65. 
 
 views respecting its transforma- 
 tion, 228. 
 Kinetic theory of gases, 172-181. 
 Knottedness in vortex rings, 201. 
 Kraft, 76, 163. 
 
 Laws of motion, 18. 
 
 Leibnitz, 76. 
 
 Le Sage, 90, 182, 238, 245. 
 
 Letters from Kelvin and Thomson, 
 
 226. 
 Light, 4, 3o. 
 Linljage of vortex rings, 200, 206. 
 
 Magnetic energy, 4, 34, 88, 136, 249. 
 Magnetism, 34. 
 
 Mass, 11, 61, 108, 147, 152, 193, 223, 
 224, 250, 252. 
 
 definition of, 11, 147. 
 
 measurement of, 148, 151. 
 
 proportional to Isinergety, 150, 
 217. 
 
 proportional to weight, 150, 217. 
 Masses, graded set of, 62, 105. 
 Materialism, 170. 
 
 Matter, 3-11, 52, 53, 61, 135-157, 185, 
 221, 230, 246, 249, 252, 
 
 conservation of, 71, 75, 156. 
 
 definition of, 10, 140, 159, 252. . 
 
 distinction between, and sub- 
 stance, 135, 147. 
 
 forms of, 140. 
 
 nature of, 140, 169, 170, 193, 195, 
 201, 227, 249. 
 
 non-resistant, 140, 250. 
 
 properties of, 142, 147, 246, 250. 
 
 quantity of, 11, 61, 147, 152. 
 
 theories of, 11, 201, 227, 249. 
 
 true, 221. 
 Maxwell, 16, 160, 175, 177, 180, 188, 
 
 220, 234. 
 Mayer, 75. 
 
 Mechanical energy, 38, 88, 89. 
 Mental effort, 22. 
 Metaphysics, 5, 140. 
 Mode of motion, 16, 232. 
 
256 
 
 INDEX. 
 
 Molecules, 155, 179, 181, 201, 202, 208, 
 252. 
 
 dimensions of, 180, 243, 245. 
 
 stability of, 214, 242. 
 Motion, 4, 12, 14, 16, 18, 19, 22. 
 
 laws of, 18. 
 
 mode of, 16, 232. 
 
 of body under force, 113, 142. 
 
 state of, 15, 18,21,40,163.' 
 
 MtJNSTERBERG, 168. 
 
 Muscular energy, 22, 36, 123. 
 
 Nerve action, 22. 
 
 Newton, 18, 73, 82, 110, 146, 163, 
 192. 
 
 Object, 6, 252. 
 Oxygen, 180. 
 
 Path, 12. 
 
 Pendulum, 101, 109. 
 
 Perpetual motion, 68, 71, 73, 74. 
 
 Perpetumn mobile, 68, 74. 
 
 Phenomena, 20, 25, 67, 151, 161. 
 
 Physical forces, 53, 67, 70. 
 
 Physical science, 6. 
 
 Position, 12. 
 
 Potential energy, 38, 126-134. 
 
 Power, 3, 25, 53, 122, 135. 
 
 Powers, 1-10, 67, 69, 123, 135, 139, 
 
 148. 
 enumeration of, 4. 
 Pressure, 41, 51. 
 Properties, 3, 7, 25, 60, 137, 142, 161, 
 
 177, 181, 193, 225, 252. 
 Psychology, 5, 140, 168, 252. 
 
 Quantivalence, 206, 211, 212, 213, 
 235, 242. 
 
 Radiant energy, 35, 76, 230, 247. 
 
 heat, 4, 3."). 
 Radiation, 4, 10, 35, 230. 
 Kankine, 126. 
 Repulsion, .50. 
 Resistance, 45, 141, 250. 
 Rest, 14. 
 
 Restraint, 16 (see Resistance) . 
 Retardation, 15, 16. 
 R-MFORD, 74, 232. 
 
 Sensation, 3, 5, 20, 23, 44, 224, 227, 
 
 251. 
 Smoke rings, 198. 
 Sound, 34. 
 Space, 3, 12. 
 
 occupancy of, by matter, 160, 225, 
 251. 
 
 rate, 96. 
 Speculations, 169. 
 Spring balance, 99. 
 Spring buffer, 55. 
 Stability of vortex rings, 207, 208. 
 Standard body, 59, 101. 
 Standards, 63, 101, 107. 
 State of motion, 15. 
 States of substances, 9. 
 Strength, 123. 
 Stress, 50. 
 
 Substance, 3-10, 135, 144, 159, 230, 
 246, 252. 
 
 conservation of, 156. 
 
 definition of, 5. 
 
 exertion of force by, 250. 
 
 nature of, 193. 
 
 properties of, 3-10, 61, 137, 142, 
 246, 249. 
 
 quantity of. 111, 147, 152. 
 System of bodies, 21. 
 
 Tait, 201. 
 
 Temperature, 178, 204, 233. 
 
 Temperature equilibrium, 178, 188, 
 190, 216. 
 
 Tension, 51. 
 
 Theory, function of, 169. 
 
 Thomson, J. J., 87, 193, 202, 204, 
 212, 226, 236. 
 
 Thomson, Sik William (see Kel- 
 vin). 
 
 Transformation of energy, 67, 78, 
 90, 117, 227. 
 
 Units, 63, 105. 
 
 Valency, 206, 211, 212, 213, 235, 242. 
 Velocity, 14. 
 Vibration, 34. 
 Vis impressa, 163. 
 Vis viva, 74, 77. 
 
INDEX. 
 
 257 
 
 Vital energy, 37. 
 Vortex atom, 202, 205. 
 
 dimensions of, 243, 245. 
 
 hypothesis, 201. 
 
 theory, 87, 90, 137, 171, 182, 193, 
 238. 
 
 tlieoi'y, crucial test of, 204. 
 
 theory, present status of, 225. 
 
 tlieory, remarks on, 219. 
 
 thickness of, 244, 245. 
 Vortex molecules, 208. 
 
 stability of, 211, 214. 
 Vortex motion, 196. 
 Vortex ring, 196, 202, 205. 
 Vortex rings, associated in rotation, 
 206, 208. 
 
 Weighing, 63, 152. 
 
 Weight, 4, 82, 100, 101, 108, 148, 152. 
 
 intensity of, 101. 
 
 proportional to kinergety, 63, 
 109, 150. 
 
 proportional to mass, 150, 217. 
 Weightal, 11, 66, 111, 152, 157. 
 
 definition of, 11, 111, 155. 
 
 proportional to kinergety, 66. 
 Weights, set of, 62, 100, 104, 111. 
 Work, 41, 76, 112-125, 160, 162, 164, 
 252. 
 
 confusion of, with energy, 118. 
 
 definition of, 117, 118, 160, 161. 
 
 done by and upon, 113, 118. 
 
 examples of, 120. 
 
 measurement of, 113, 118. 
 
 performance of, 118. 
 
 Young, 77. 
 
BY THE AUTHOR. 
 
 PuBuSHED BY THE MACMILLAN COMPANY, 66 Fifth Avenue, New York. 
 
 MATTER, ENERGY, FORCE, AND WORK. 
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 2