Outlines of the Science of Energetics

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Outlines of the Science of Energetics  (1855) 
by William John Macquorn Rankine
In: Proceedings of the Royal Philosophical Society of Glasgow, Volume 3, by Royal Philosophical Society of Glasgow, 1855. pp.121-141.

May 2, 1855 (the Concluding Meeting of the Session was held this evening) William Gourlie, Esq., Vice-President, in the Chair.
Mr. J. Namer read a paper " On the Chemistry of Trap Dykes in Arran."
Mr. W. J. Macquorn Rankine read a paper " On the Science of Energetics."

What Constitutes A Physical Theory.[edit]

An essential distinction exists between two stages in the process of advancing our knowledge of the laws of physical phenomena; the first stage consists in observing the relations of phenomena, whether of such as occur in the ordinary course of nature, or of such as are artificially produced in experimental investigations, and in expressing the relations so observed by propositions called formal laws. The second stage consists in reducing the formal laws of an entire class of phenomena to the form of a science; that is to say, in discovering the most simple system of principles, from which all the formal laws of the class of phenomena can be deduced as consequences.

Such a system of principles, with its consequences methodically deduced, constitutes the physical theory of a class of phenomena.

A physical theory, like an abstract science, consists of definitions and axioms as first principles, and of propositions, their consequences; but with these differences:—first, That in an abstract science, a definition assigns a name to a class of notions derived originally from observation, but not necessarily corresponding to any existing objects of real phenomena, and an axiom states a mutual relation amongst such notions, or the names denoting them; while in a physical science, a definition states properties common to a class of existing objects, or real phenomena, and a physical axiom states a general law as to the relations of phenomena; and, secondly,—That in an abstract science, the propositions first discovered are the most simple; whilst in a physical theory, the propositions first discovered are in general numerous and complex, being formal laws, the immediate results of observation and experiment, from which the definitions and axioms are subsequently arrived at by a process of reasoning differing from that whereby one proposition is deduced from another in an abstract science, partly in being more complex and difficult, and partly in being to a certain extent tentative, that is to say, involving the trial of conjectural principles, and their acceptance or rejection according as their consequences are found to agree or disagree with the formal laws deduced immediately from observation and experiment.

The Abstractive Method of Forming a Physical Theory, Distinguished from the Hypothetical Method.[edit]

Two methods of framing a physical theory may be distinguished, characterized chiefly by the manner in which classes of phenomena are defined. They may be termed respectively the abstractive and the hypothetical methods.

According to the abstractive method, a class of objects or phenomena is defined by describing, or otherwise making to be understood, and assigning a name or symbol to, that assemblage of properties which is common to all the objects or phenomena composing the class, as perceived by the senses, without introducing anything hypothetical.

According to the hypothetical method, a class of objects or phenomena is defined according to a conjectural conception of their nature, as being constituted in a manner not apparent to the senses, by a modification of some other class of objects or phenomena whose laws are already known. Should the consequences of such a hypothetical definition be found to be in accordance with the results of observation and experiment, it serves as the means of deducing the laws of one class of objects or phenomena from those of another.

The conjectural conceptions involved in the hypothetical method may be distinguished into two classes, according as they are adopted as a probable representation of a state of things which may really exist, though imperceptible to the senses, or merely as a convenient means of expressing the laws of phenomena; two kinds of hypotheses, of which the former may be called objective, and the latter subjective. As examples of objective hypotheses may be taken, that of vibrations or oscillations in the theory of light, and that of atoms in chemistry; as an example of a subjective hypothesis, that of magnetic fluids.

The Science or Mechanics Considered as an Illustration of the Abstractive Method.[edit]

The principles of the science of mechanics, the only example yet existing of a complete physical theory, are altogether formed from the data of experience by the abstractive method. The class of objects to which the science of mechanics relates,—viz.,—material bodies,—are defined by means of those sensible properties which they all possess, viz., the property of occupying space, and that of resisting change of motion. The two classes of phenomena to which the science of mechanics relates are distinguished by two words, motion and force; motion being a word denoting that which is common to the fall of heavy bodies, the flow of streams, the tides, the winds, the vibrations of sonorous bodies, the revolutions of the stars, and generally to all phenomena involving change of the portions of space occupied by bodies; and force, a word denoting that which is common to the mutual attractions and repulsions of bodies, distant or near, and of the parts of bodies, the mutual pressure or stress of bodies in contact, and of the parts of bodies, the muscular exertions of animals, and, generally, to all phenomena tending to produce or to prevent motion.

The laws of the composition and resolution of motions, and of the composition and resolution of forces, are expressed by propositions which are the consequences of the definitions of motion and force respectively. The laws of the relations between motion and force are the consequences of certain axioms, being the most simple and general expressions for all that has been ascertained by experience respecting those relations.

Mechanical Hypotheses In Various Branches or Physics.[edit]

The fact that the theory of motions and motive forces is the only complete physical theory, has naturally led to the adoption of mechanical hypotheses in the theories of other branches of physics; that is to say, hypothetical definitions, in which classes of phenomena are defined conjecturally as being constituted by some kind of motion or motive force not obvious to the senses (called molecular motion or force) as when light and radiant heat as defined as consisting in molecular vibrations, thermometric heat in molecular vortices, and the rigidity of solids in molecular attractions and repulsions.

The hypothetical motions and forces are sometimes ascribed to hypothetical bodies, such as the luminiferous aether; sometimes to hypothetical parts, whereof tangible bodies are conjecturally defined to consist, such as atoms, atomic nuclei with elastic atmospheres, and the like.

A mechanical hypothesis is held to have fulfilled its object, when, by applying the known axioms of mechanics to the hypothetical motions and forces, results are obtained agreeing with the observed laws of the classes of phenomena under consideration, and when, by the aid of such a hypothesis, phenomena previously unobserved are predicted, and laws anticipated, it attains a high degree of probability.

A mechanical hypothesis is the better, the more extensive the range of phenomena whose laws it serves to deduce from the axioms of mechanics; and the perfection of such a hypothesis would be, if it could, by means of one connected system of suppositions, be made to form a basis for all branches of molecular physics.

Advantages and Disadvantages of Hypothetical Theories.[edit]

It is well known that certain hypothetical theories, such as the wave theory of light, have proved extremely useful, by reducing the laws of a various and complicated class of phenomena to a few simple principles, and by anticipating laws afterwards verified by observation.

Such are the results to be expected from well-framed hypotheses in every branch of physics, when used with judgment, and especially with that caution which arises from the consideration, that even those hypotheses whose consequences are most fully confirmed by experiment, never can by any amount of evidence attain that degree of certainty which belongs to observed facts.

Of mechanical hypotheses in particular, it is to be observed, that their tendency is to combine all branches of physics into one system, by making the axioms of mechanics the first principles of the laws of all phenomena; an object for the attainment of which an earnest wish was expressed by Newton.[1]

In the mechanical theories of elasticity, light, heat, and electricity, considerable progress has been made towards that end.

The neglect of the caution already referred to, however, has caused some hypotheses to assume, in the minds of the public generally, as well as in those of many scientific men, that authority which belongs to facts alone, and a tendency has consequently often evinced itself to explain away, or set aside, facts inconsistent with these hypotheses, which facts, rightly appreciated, would have formed the basis of true theories; thus the fact of the production of heat by friction, the basis of the true theory of heat, was long neglected, because inconsistent with the hypothesis of caloric; and the fact of the production of cold by electric currents, at certain metallic junctions, the key (as Professor William Thomson recently showed) to the true theory of the phenomena of thermo-electricity, was, from inconsistency with prevalent assumptions respecting the so-called "electric fluid," by some regarded as a thing to be explained away, and by others as a delusion.

Such are the evils which arise from the misuse of hypothesis.

Advantages of an Extension of the Abstractive Method of Framing Theories.[edit]

Besides the perfecting of Mechanical Hypotheses, another and an entirely distinct method presents itself for combining the physical sciences into one system; and that it is by an extension of the Abstractive Process in framing Theories.

The abstractive method has already been partially applied, and with success, to special branches of molecular physics, such as heat, electricity, and magnetism. We are now to consider in what manner it is to be applied to physics generally, considered as one science.

Instead of supposing the various classes of physical phenomena to be constituted in an occult way of modifications of motion and force, let us distinguish the properties which those classes possess in common with each other, and so define more extensive classes denoted by suitable terms. For axioms, to express the laws of those more extensive classes of phenomena, let us frame propositions comprehending as particular cases, the laws of the particular classes of phenomena comprehended under the more extensive classes. So shall we arrive at a body of principles, applicable to physical phenomena in general, and which being framed by induction from facts alone, will be free from the uncertainty which must always attach even to those mechanical hypotheses whose consequences are most fully confirmed by experiment. This extension of the abstractive process is not proposed in order to supersede the hypothetical method of theorizing; for in almost every branch of molecular physics it may be held, that a hypothetical theory is necessary as a preliminary step to reduce the expression of the phenomena to simplicity and order, before it is possible to make any progress in framing an abstractive theory.

Nature of the Science of Energetics.[edit]

Energy, or the capacity to effect changes, is the common characteristic of the various states of matter to which the several branches of physics relate; if, then, there be general laws respecting energy, such laws must be applicable, mutatis mutandis, to every branch of physics, and must express a body of principles as to physical phenomena in general.

In a paper read to the Philosophical Society of Glasgow on the 5th of January 1853, a first attempt was made to investigate such principles, by defining actual energy and potential energy, and by demonstrating a general law of the mutual transformations of those kinds of energy, of which one particular case is a previously known law of the mechanical action of heat in elastic bodies, and another, a subsequently demonstrated law which forms the basis of Professor William Thomson's Theory of thermo-electricity.

The object of the present paper is, to present in a more systematic form, both these and some other principles, forming part of a science whose subjects are, material bodies and physical phenomena in general, and which it is proposed to call the Science Of Energetics.

Definitions of Certain Terms.[edit]

The peculiar terms which will be used in treating of the Science of Energetics are purely abstract; that is to say, they are not the names of any particular object, nor of any particular phenomena, nor of any particular notions of the mind, but are names of very comprehensive classes of objects and phenomena. About such classes it is impossible to think or to reason, except by the aid of examples or of symbols. General terms are symbols employed for this purpose.


The term "substance" will be applied to all bodies, parts of bodies, and systems of bodies. The parts of a substance may be spoken of as distinct substances, and a system of substances related to each other may be spoken of as one complex substance. Strictly speaking, the term should be "material substance," but it is easily borne in mind, that in this essay none but material substances are referred to.


The term "property" will be restricted to invariable properties; whether such as always belong to all material substances, or such as constitute the invariable distinctions between one kind of substance and another.


Mass means "quantity of substance." Masses of one kind of substance may be compared together by ascertaining the numbers of equal parts which they contain; masses of substances of different kinds are compared by means to be afterwards referred to.


The term "accident" will be applied to every variable state of substances, whether consisting in a condition of each part of a substance, how small soever, (which may be called an absolute accident), or in a physical relation between parts of substances, (which may be called a relative accident). Accidents to be the subject of scientific inquiry, must be capable of being measured and expressed by means of quantities. The quantity, even of an absolute accident, can only be expressed by means of a mentally-conceived relation.

The whole condition or state of a substance, so far as it is variable, is a complex accident; the independent quantities which are at once necessary and sufficient to express completely this complex accident, are independent accidents. To express the same complex accident, different systems of independent accidents may be employed; but the number of independent accidents in each system will be the same.

Examples.— The variable thermic condition of an elastic fluid is a complex accident, capable of being completely expressed by two independent accidents, which may be any two out of these three quantities—the temperature, the density, the pressure—or any two independent functions of these quantities.

The condition of strain at a point in an elastic solid, is a complex accident, capable of being completely expressed by six independent accidents, which may be the three elongations of the dimensions, and the three distortions of the faces of a molecule originally cubical, or the lengths and directions of the axes of the ellipsoidal figure assumed by a molecule originally spherical; or any six independent functions of either of those systems of quantities.

The distinction of accidents into absolute and relative is to a certain extent arbitrary; thus, the figure and dimensions of a molecule may be regarded as absolute accidents, when it is considered as a whole, or as relative accidents, when it is considered as made up of parts. Most kinds of accidents are necessarily relative, but some kinds can only be considered as relative accidents when some hypothesis is adopted as to the occult condition of the substances which they affect, as when heat is ascribed hypothetically to molecular motions; and such suppositions are excluded from the present inquiry.

Accidents may be said to be homogeneous when the quantities expressing them are capable of being put together, so that the result of the combination of the different accidents shall be expressed by one quantity. The number of heterogeneous kinds of accidents is evidently indefinite.

Effort, or Active Accident.[edit]

The term "effort" will be applied to every cause which varies, or tends to vary, an accident. This term, therefore, comprehends not merely forces or pressures, to which it is usually applied, but all causes of variation in the condition of substances.

Efforts may be homogeneous or heterogeneous.

Homogeneous efforts are compared by balancing them against each other.

An effort, being a condition of the parts of a substance, or a relation between substances, is itself an accident, and may be distinguished as an "active accident."

With reference to a given limited substance, internal efforts are those which consist in actions amongst its parts; external efforts those which consist in actions between the given substance and other substances.

Passive Accident.[edit]

The condition which an effort tends to vary may be called a "passive accident," and when the word "accident" is not otherwise qualified, "passive accident" may be understood.

Radical Accident.[edit]

If there be a quantity such that it expresses at once the magnitude of the passive accident caused by a given effort, and the magnitude of the active accident or effort itself, let the condition denoted by that quantity be called a "radical accident."

[The velocity of a given mass is an example of a radical accident, for it is itself a passive accident, and also the measure of the kind of effort called accelerative force, which acting for unity of time, is capable of producing that passive accident.]

[The strength of an electric current is also a radical accident.]

Effort as a Measure of Mass.[edit]

Masses, whether homogeneous or heterogeneous, may be compared by means of the efforts required to produce in them variations of some particular accident. The accident conventionally employed for this purpose is velocity.


"Work" is the variation of an accident by an effort, and is a term comprehending all phenomena in which physical change takes place. Quantity of work is measured by the product of the variation of the passive accident by the magnitude of the effort, when this is constant; or by the integral of the effort, with respect to the passive accident, when the effort is variable.

Let x denote a passive accident.

X an effort tending to vary it.

W the work performed in increasing x from x_0 to x_1, then,

W=\int_{x_0}^{x_1}X\,dx\mathrm{,\ and}&\\
W=X(x_1-x_0)\mathrm{\ if\ X\ is\ constant.}&\end{cases} 


 ( 1)


Work is represented geometrically by the area of a curve, whereof the abscissa represents the passive accident, and the ordinate, the effort.

Energy, Actual and Potential.[edit]

The term "energy" comprehends every state of a substance which constitutes a capacity for performing work. Quantities of energy are measured by the quantities of work which they constitute the means of performing.

"Actual energy" comprehends those kinds of capacity for performing work which consist in particular states of each part of a substance, how small soever; that is, in an absolute accident, such as heat, light, electric current, vis-viva. Actual energy is essentially positive.

"Potential energy" comprehends those kinds of capacity for performing work which consist in relations between substances, or parts of substances; that is, in relative accidents. To constitute potential energy there must be a passive accident capable of variation, and an effort tending to produce such variation; the integral of this effort, with respect to the possible variation of the passive accident, is potential energy, which differs in work from this—that in work the change has been effected, which, in potential energy, is capable of being effected.

Let x denote an accident, x_1 its actual value; X, an effort tending to vary it; x_Q, the value to which the effort tends to bring the accident; then

\int_{x_1}^{x_0}X\,dx = U, denotes potential energy.

Examples of potential energy are, the chemical affinity of uncombined elements; the energy of gravitation, of magnetism, of electrical attraction and repulsion, of electro-motive force, of that part of elasticity which arises from actions between the parts of a body, and generally, of all mutual actions of bodies, and parts of bodies.

Potential energy may be passive or negative, according as the effort in question is of the same sign with the variation of the passive accident, or of the opposite sign; that is, according as X is of the same sign with dx, or of the opposite sign.

It is to be observed, that the states of substances comprehended under the term actual energy, may possess the characteristics of potential energy also; that is to say, may be accompanied by a tendency or effort to vary relative accidents; as heat, in an elastic fluid, is accompanied by a tendency to expand ; that is, an effort to increase the volume of the receptacle containing the fluid.

The states to which the term, potential energy, is especially applied, are those which are solely due to mutual actions.

To put a substance into a state of energy, or to increase its energy, is obviously a kind of work.

First Axiom: All kinds of Work and Energy are Homogeneous.[edit]

This axiom means, that any kind of energy may be made the means of performing any kind of work. It is a fact arrived at by induction from experiment and observation, and its establishment is more especially due to the experiments of M. Joule.

This axiom leads, in many respects, to the same consequences with the hypothesis that all those kinds of energy which are not sensibly the results of motion and motive force are the results of occult modifications of motion and motive force.

But the axiom differs from the hypothesis in this, that the axiom is simply the generalized allegation of the facts proved by experience, while the hypothesis involves conjectures as to objects and phenomena which never can be subjected to observation.

It is the truth of this axiom which renders a science of energetics possible.

The efforts and passive accidents to which the branches of physics relate are varied and heterogeneous; but they are all connected with energy, a uniform species of quantity, which pervades every branch of physics.

This axiom is also equivalent to saying, that energy is transformable and transferable (an allegation which, in the previous paper referred to, was included in the definition of energy): for, to transform energy, means to employ energy depending on accidents of one kind, in putting a substance into a state of energy depending on accidents of another kind ; and to transfer energy, means to employ the energy of one substance in putting another substance into a state of energy, both of which are kinds of work, and may, according to the axiom, be performed by means of any kind of energy.

Second Axiom. : The Total Energy of a Substance Cannot be Altered by the Mutual Actions of its Parts.[edit]

Of the truth of this axiom there can be no doubt; but some difference of opinion may exist as to the evidence on which it rests. There is ample experimental evidence from which it might be proved; but independently of such evidence, there is the argument, that the law expressed by this axiom is essential to the stability of the universe, such as it exists. The special application of this law to mechanics is expressed in two ways, which are virtually equivalent to each other; the principle of visviva, and that of the equality of action and reaction. The latter principle is demonstrated by Newton, from considerations connected with the stability of the universe (Principia, Scholium to the Laws of Motion) ; for he shows, that but for the equality of action and reaction, the earth, with a continually accelerated velocity, would fly away through infinite space.

It follows, from the Second Axiom, that all work consists in the transfer and transformation of energy alone; for otherwise the total amount of energy would be altered. Also, that the energy of a substance can be varied by external efforts alone.

External Potential Equilirrium.[edit]

The entire condition of a substance, so far as it is variable, as explained in Article VIII., under the head of accident, is a complex accident, which may be expressed in various ways by means of different systems of quantities denoting independent accidents; but the number of independent accidents in each system must be the same.

The quantity of work required to produce any change in the condition of the substance, that is to say, the potential energy received by it from without, during that change, may in like manner be expressed in different ways by the sums of different systems of integrals of external efforts, each integrated with respect to the independent accident which it tends to augment; but the number of integrals in each system, and the number of efforts, like the number of independent accidents, must be the same; and so also must the sums of the integrals, each sum representing the same quantity of work in a different way.

The different systems of efforts which correspond to different systems of independent accidents, each expressing the same complex accident, may be called equivalent systems of efforts; and the finding of a system of efforts equivalent to another may be called conversion of efforts.*

  • The conversion of efforts in Physics, is connected with the theory of lineal transformations in Algebra.

When the law of variation of potential energy, by a change of condition of a substance, is known, the system of external efforts corresponding to any system of independent accidents is found by means of this principle:

Each effort is equal to the rate of variation of the potential energy tvith respect to the independent accident which that effort tends to vary; or symbolically

External Potential Equilirrium of a substance takes place, when the external effort to vary each of the independent accidents is null; that is to say, when the rate of variation of the potential energy of the substance with the variation of each independent accident is null.

For a given substance, there are as many conditions of equilibrium, of the form dTJ

(3.) -5-- o,

as there are independent accidents in the expression of its condition. The special application of this law to motion and motive force constitutes the principle of virtual velocities, from which the whole science of statics is deducible.

Internal Potential Equilirrium.[edit]

The internal potential equilibrium of a substance consists in the equilibrium of each of its parts, considered separately ; that is to say, in the nullity of the rate of variation of the potential energy of each part with respect to each of the independent accidents on which the condition of such part depends.

Examples of particular cases of this principle are, the laws of the equilibrium of elastic solids, and of the distribution of statical electricity.

Third Axiom.[edit]

The Effort to Perform Work of a Given Kind, Caused by a Given Quantity of Actual Energy, is the Sum of the Efforts Caused by the Parts of thai Quantity.

A law equivalent to this axiom, under the name of the " General Law Of The Transformation Of Energy," formed the principal subject of the previous paper already referred to.

This axiom appears to be a consequence of the definition of actual energy, as a capacity for performing work possessed by each part of a substance independently of its relations to other parts, rather than an independent proposition.

Its applicability to natural phenomena arises from the fact, that there are states of substances corresponding to the definition of actual energy.

The mode of applying this third axiom is as follows:—

Let a homogeneous substance possess a quantity Q, of a particular kind of actual energy, uniformly distributed, and let it be required to determine the amount of the effort arising from the actual energy, which tends to perform a particular kind of work W, by the variation of a particular passive accident x.

The total effort to perform this kind of work is represented by the rate of its increase relatively to the passive accident, viz.,— X= — dx

Divide the quantity of actual energy Q into an indefinite number of indefinitely small parts 3Q ; the portion of the effort X due to each of those parts will be dX -and adding these partial efforts together, the effort caused by the whole quantity of actual energy will be

(4.) Q^-=Qd2W dQ — dQ dx

If this be equal to the effective effort X, then that effort is simply proportional to, and wholly caused by, the actual energy Q. This is the case of the pressure of a perfect gas, and the centrifugal force of a moving body. If the effort caused by the actual energy differs from the effective effort, their difference represents, when the former is the less, an additional effort dQ

(5.) And when the former is the greater, a counter effort due to some other cause or causes.

Rate Of Transformation ; Metamorphic Function.[edit]

The effort to augment a given accident x, caused by actual energy of a given kind Q, may also be called the "Hate of Transformation" of the given kind of actual energy with increase of the given accident; for the limit of the amount of actual energy which disappears in performing work by an indefinitely small augmentation dx, of the accident, is

(6-) ^ aQ = Q*JL dX=QdTM dQdx dQ

The last form of the above expression is obviously applicable when the work -W is the result of the variation of any number of independent accidents, each by the corresponding effort. For example, let x, y, z, &c., be any number of independent accidents, and X, Y, Z, &c., the efforts to augment them ; so that dW = Xdx + Ydy + Zcfe + &c. Then (7.) rfH-Q^ + g* +%* + *»' | = Qd ■ as before. dQ The function of actual energy, efforts, and passive accidents, denoted by dW— pdH. dQ

whose variation, multiplied by the actual energy, gives the amount of actual energy transformed in performing the work d W, may be called the " Metamorphic Function" of the kind of actual energy Q relatively to the kind of work W.

When this metamorphic function is known for a given homogeneous substance, the quantity H of actual energy of the kind Q transformed to the kind of work W, during a given operation, is found by taking the integral

(9.) H= /*QdF.

The transformation of actual energy into work by the variation of passive accidents is a reversible operation; that is to say, if the passive accidents be made to vary to an equal extent in an opposite direction, potential energy will be exerted upon the substance, and transformed into actual energy: a case represented by the expression (9.) becoming negative.

The metamorphic function of heat relatively to expansive power, was first employed in a paper on the Economy of Heat in Expansive Machines, read to the Royal Society of Edinburgh in April 1851 ("Trans. Roy. Soc. Edin.," vol. xxi.)

The metamorphic function of heat relatively to electricity was employed by Professor William Thomson, in a paper on Thermo-Electricity, read to the Royal Society of Edinburgh in May 1854 ("Trans. Roy. Soc. Edin.," vol. xxi.), and was the means of anticipating some most remarkable laws, afterwards confirmed by experiment.

Equilirrium Of Actual Energy ; Metabatic Function.[edit]

It is known by experiment, that a state of actual energy is directly transferable; that is to say, the actual energy of a particular kind (such as heat), in one substance, may be diminished, the sole work performed being an equal augmentation of the same kind of actual energy in another substance.

Equilibrium of actual energy of a particular kind Q between substances A and B, takes place, when the tendency of B to transfer this kind of energy to B is equal to the tendency of B to transfer the same kind of energy to A.

Laws respecting the equilibrium of particular kinds of actual energy have been ascertained by experiment, and in some cases anticipated by means of mechanical hypotheses, according to which, all actual energy consists in the vis-viva of motion.

The following law will now be proved, respecting the equilibrium of actual energy of all possible kinds :— Theorem.—If Equilirrium Of Actual Energy Of A Given Kind


Demonstration.—The tendency of one substance to transfer actual energy of the kind Q to another, must depend on some sort of effort, whose nature and laws may be known or unknown. Let YA be this effort for the substance A, YB the corresponding effort for the substance B. Then a condition of equilibrium of actual energy is

(10.) YA = Y,

The effort Y may or may not be proportionate to the actual energy Q multiplied by a quantity independent of Q. Case first.—If it is so proportional, let Y== Iq, K being independent of Q ; then the condition of equilibrium becomes Q.=K. a ratio independent of the absolute amounts of actual energy.

Case second.—If the effort Y is not simply proportional to the actual energy Q, the portion of it caused by that actual energy, according to the principle of article 13, deduced from the third axiom, is, for each substance, and a second condition of equilibrium of actual energy is furnished by the equation

In order that this condition may be fulfilled simultaneously with the condition (10.) it is necessary that that is to say, that the ratio of the quantities of actual energy in the two substances should be independent of those quantities themselves; a condition expressed, as before, by

(11.) Jk=ik v ; Q. KA Q.E.D.

This ratio is a quantity to be ascertained by experiment, and may be called the ratio of the Specific Actual Energies of the substances A and B, for the kind of energy under consideration. The function

(12) ^.A_= S» == t

whose identity for the two substances expresses the condition of equilibrium of the actual energy Q between them, may be called the " MetaBatic Function" for that kind of energy.

In the science of thermo-dynamics, the metabatic function is absolute temperature; and the factor K is real specific heat. The theorem stated above, when applied to heat, amounts to this: that the real specific heat of a substance is independent of its temperature.

Use Of The Metabatic Function; Transformation Of Energy In An Aggregate.[edit]

From the mutual proportionality of the actual energy Q, and the metabatic function 4, it follows that the operations Qjl, 6J dQ de are equivalent; and that the latter may be substituted for the former in all the equations expressing the laws of the transformation of energy. We have therefore

(13.) QJ«L=*J^=^ dQ d0 d6dx

for the effort to transform actual energy of the kind Q into work of the kind W, when expressed in terms of the metabatic function; and

(14.) dK = dd—

for the limit of the indefinitely small transformation produced by an indefinitely small variation of the accidents on which the kind of work W depends.

There is also a form of metamorphic function.

(15.) *-*!L-/\£UKF dt J 6 suited for employment along with the metabatic function, in order to find, by the integration

(16.) H=3/<d»

the quantity of actual energy of a given kind Q transformed to the kind of work W during any finite variation of accidents.

The advantage of the above expressions is, that they are applicable, not merely to a homogeneous substance, but to any heterogeneous substance or aggregate, which is internally in a state of equilibrium of actual and potential energy; for throughout all the parts of an aggregate in that condition, the metabatic function t is the same, and each of the efforts X, &c., is the same, and consequently the metamorphic function <p is the same.

" Carnot's function" in thermo-dynamics is proportional to the reciprocal of the metabatic function of heat.

Efficiency Of Engines.[edit]

An engine is a contrivance for transforming energy by means of the periodical repetition of a cycle of variations of the accidents of a substance.

The efficiency of an engine is the proportion which the energy permanently transformed to a useful form by it bears to the whole energy communicated to the working substance.

In a perfect engine the cycle of variations is thus:—

I. The metabatic function is increased, say from do to h.
II. The metamorphic function is increased by the amount A <p.
III. The metabatic function is diminished from $1 back to $0.
IV. The metamorphic function is diminished by the amount A <p.

During the second operation, the energy received by the working substance, and transformed from the actual to the potential form is dx A <p. During the fourth operation energy is transformed back, to the amount i0 A <p. So that the energy permanently transformed during each cycle A a is (0!—io) A <p; and the efficiency of the engine-! °

Diffusion Of Actual Energy ; Irreversible Or Frictional Operations.[edit]

There is a tendency in every substance or system of substances, to the equable diffusion of actual energy ; that is to say, to its transfer between the parts of the substance or system, until the value of the metabatic function becomes uniform.

This process is not directly reversible; that is to say, there is no such operation as a direct concentration of actual energy through a tendency of the metabatic function to become unequal in different parts of a substance or system.

Hence arises the impossibility of using the energy re-converted to the actual form at the lower limit of the metabatic function in an engine.

There is an analogy in respect of this property of irreversibility, between the diffusion of one kind of actual energy, and certain irreversible transformations of one kind of actual energy to another, called by Professor William Thomson, "Frictional phenomena," viz., the production of heat by rubbing and agitation, and by electric currents in a homogeneous substance at a uniform temperature.

In fact, a conjecture may be hazarded, that immediate diffusion of the actual energy produced in frictional phenomena, is the circumstance which renders them irreversible ; for, suppose a small part of a substance to have its actual energy increased by the exertion of some kind of work upon it; then, if the increase of actual energy so produced be immediately diffused amongst other parts, so as to restore the uniformity of the metabatic function, the whole process will be irreversible. This speculation, however, is, for the present, partly hypothetical; and, therefore does not, strictly speaking, form part of the science of energetics.

Measurement Of Time.[edit]

The general relations between energy and time must form an important branch of the science of energetics; but for the present, all that I am prepared to state on this subject is the following Definition Of Equal Times:

Equal times are the times in which equal quantities of the same hind of work are 'performed by equal and similar substances, under wholly similar circumstances.

Concluding Remarks.[edit]

It is to be observed, that the preceding articles are not the results of a new and hitherto untried speculation, but are the generalized expression of a method of reasoning which has already been applied with success to special branches of physics.

In this brief essay, it has not been attempted to do more than to give an outline of some of the more obvious principles of the science of energetics, or the abstract theory of physical phenomena in general; a science to which the maxim, true of all science, is specially applicable— that its subjects are boundless, and that they never can, by human labours, be exhausted, nor the science brought to perfection.


  1. Utinam cætera naturæ phænomena ex principiis mechanicis eodem argumentandi genere derivare liceret.—(Phil. Nat. Prin. Math.; Præf.)

This work published before January 1, 1923 is in the public domain worldwide because the author died at least 100 years ago.