ENERGY may be defined as the power of doing work, or of overcoming resistance. A bent spring possesses energy, for it is capable of doing work in returning to its natural form ; a charge of gunpowder possesses energy, for it is capable of doing work in exploding; a Leyden jar charged with electricity possesses energy, for it is capable of doing work in being discharged. A complete account of our knowledge of energy and its transformations would require an exhaustive treatise on every branch of physical science, for natural philosophy is simply the science of energy. There are, however, certain general principles to which energy conforms in all the varied transformations which it is capable of undergoing, and of these principles we propose to give a brief sketch.

Before we can treat energy as a physical quantity we must possess some means of measuring it. If we raise 1 lb of matter through a foot we do a certain amount of work against the earth's attraction. If we raise 2 S> through the same height we do twice this amount of work, and so on for any number of pounds, so that the work done is proportional to the mass raised, and therefore to the resist-ance overcome. Also, if we neglect the variation of the intensity of gravity, the work done in raising 1 lb through 2 feet will be double of that done in raising it 1 foot. Hence we conclude that the work done varies as the resist-ance overcome and the distance through which it is overcome conjointly.

Now, we may select any definite quantity of work we please as our unit, as, for example, the work done in lifting a pound a foot high from the sea-level in the latitude of "London, which is the unit of work generally adopted by British engineers, and is called the "foot-pound." The most useful unit for scientific purposes is one which depends only on the fundamental units of length, mass, and time, and is hence called an absolute unit. Such a unit is independent of gravity or of any other quantity which varies with the locality. Taking the centimetre, gramme, and second as our fundamental units, the most convenient unit of force is that which, acting on a gramme for a second, produces in it a velocity of a centimetre per second; this is called a Dyne. The unit of work is that which is re-quired to overcome a resistance of a dyne over a centimetre, and is called an Erg. In the latitude of Paris the dyne is equal to the weight of about -g-L of a gramme, and the erg is the amount of work required to raise -g~g-T of a gramme vertically through one centimetre. A megalerg is one million ergs.

Energy is the capacity for doing work. The unit of energy should therefore be the same as that of work, and the centimetre-gramme-second (or, as it is usually called, the C.G.S.) unit of energy is the erg.

The forms of energy which are most readily recognized are of course those in which the energy can be most readily employed in doing mechanical work, and it is manifest that masses of matter which are large enough to be seen and handled are more readily dealt with mechanically than are smaller masses. Hence when useful work can be obtained from a system by simply connecting visible portions of it by a train of mechanism, such energy is more readily recognized than is that which compels us to control the behaviour of molecules before we can transform it into useful work. The former is sometimes, though very improperly, called visible energy, because its transformation is always accompanied by a visible change in the system itself.

The conception of work and of energy was originally derived from observation of purely mechanical phenomena, that is to say, phenomena in which the relative positions and motions of visible portions of matter were all that were taken into consideration. Hence it is not surprising that, in those more subtle forms in which energy cannot be so readily converted into work, it should for a long while have escaped recognition after it had become familiar to the student of dynamics.

If a pound weight be suspended by a string passing over a pulley, in descending through 10 feet it is capable of raising nearly a pound weight, attached to the other end of the string, through the same height, and thus can do nearly 10 foot-pounds of work. The smoother we make the pulley the more nearly does the amount of useful work which the weight is capable of doing approach 10 foot-pounds, and if we take into account the work done against the friction of the pulley, we may say that the work done by the descending weight is 10 foot-pounds, and hence when the weight is in its elevated position we have at disposal 10' foot-pounds more energy than when it is in the lower posi-tion. It should be noticed, however, that this energy is possessed by the system consisting of the earth and pound together, in virtue of their separation, and that neither could do work without the other to attract it. The system consisting of the earth and the pound therefore possesses an amount of energy which depends on the relative positions of its two parts, and the stresses existing between them. In most mechanical systems the stresses acting between the parts can be determined when the relative positions of all the parts are known ; and the energy which a system possesses in virtue of the relative positions of its parts, or its configuration, is called its " Potential Energy," to distinguish it from another form of energy which we shall presently consider. The word potential does not imply that this energy is not real and exists only in potentiality; it is energy, and has as much claim to the title as it has in any other form in which it may appear.

It is a well-known proposition in dynamics that, if a body be projected vertically upwards in vacuo, with a velocity of v centimetres per second, it will rise to a height

„2
of — centimetres, where g represents the numerical value

of the acceleration produced by gravity in centimetre-second units. Now, if m represent the mass of the body in grammes, its weight will be mg dynes, for it will require a. force of mg dynes to produce in it the acceleration denoted by g. Hence the work done in raising the mass will be represented by mg—, that is, Jmt>2 ergs. But it is merely

_i . .

in virtue of the velocity of projection that the mass is capable of rising against the resistance of gravity, and hence we must conclude that at the instant of projec-tion it possessed \mvL units of energy. Now, whatever be the direction in which a body is moving, a friction-less constraint, like a string attached to the body, can cause its velocity to be changed into the vertical direc-tion without any change taking place in the magnitude of the velocity. Hence we may say that if a body of mass m be moving in any direction relative to the earth, we have at disposal, in virtue of this motion, \mv2 units of energy, and this is converted into potential energy if the body come to rest at the highest point of its path. Like potential energy, this energy is relative and is due to the motion of the body relative to the earth, for we know nothing about absolute motion in space; and, moreover,, when we have brought the body to rest relative to the earth, we shall have deprived it of all the energy which we can derive from its motion. The energy is there-fore possessed in common by the system consisting of the earth and the body; and the energy which a system pos-sesses in virtue of the relative motions of its parts is called "Kinetic Energy."

A good example of the transformation of kinetic energy into potential energy, and vice versa, is seen in the pendu-lum. When at the limits of its swing, the pendulum is for an instant at rest, and all the energy of the oscillation is potential. When passing through its position of equilibrium, since gravity can do no more work upon it without changing its fixed point of support, all the energy of oscillation is kinetic. At intermediate positions the energy is partly kinetic and partly potential.

Kinetic energy is possessed by a system of two or more bodies in virtue of the relative motion of its parts. Since our conception of velocity is essentially relative, and we know nothing about absolute velocities in space, it is plain that any property possessed by a body in virtue of its motion can be possessed by it only in relation to those bodies with respect to which it is moving, and thus a single rigid body can never be said to possess kinetic energy in virtue of the motion of its centre of mass. If a body whose mass is m grammes be moving with a velocity of v centimetres per second relative to the earth, the kinetic energy possessed by the system is -|TOW2 ergs if m be small relative to the earth. But if we consider two bodies each of mass m and one of them moving with velocity v rela-tive to the other, we can only obtain \mv- units of work from this system alone, and we ought not to say that the system considered by itself possesses more than \mvl units of energy. If we include the earth in our system the whole energy will depend on the velocities of the bodies relative to the earth, and not simply on their velocities relative to one another. Hence whenever we say that the kinetic energy of a body is ijmv2, we mean its kinetic energy relative to the earth, and the statement is only true when the mass of the body is very small compared with that of the earth. Any general expression for the energy of a system ought to be true whatever body in the system we consider fixed. It is manifest that the expression \mvi will not be a true re-presentation of the kinetic energy of the earth and a cannon shot if we choose to consider the shot fixed and the earth moving towards it. In fact any general expression for the energy of a system must involve the masses of all the bodies concerned; but if the mass of one body be infinite compared with that of any of the others we may adopt the expression |-2(mi>2) for the kinetic energy, the body of infinite mass being supposed at rest.

It is only when a body possesses no motion of rotation that we may speak of its velocity as a whole. If a body be rotating about an axis, it follows from D'Alembert's principle that the work it is capable of doing while being brought to rest is the same as if each particle were perfectly free and moving with the velocity which it actually possesses. Hence if the moment of inertia of a body about its axis of rotation be represented by I, and its angular velocity by u), the work which can be done by it if we can succeed in bringing it to rest will be |Io>2. We shall see hereafter how this energy may be transformed without the help of any external body if we suppose the rotating body indefinitely extensible in any direction at right angles to the axis of rotation, so that there is a sense in which we may speak of the kinetic energy of rotation as really belonging to the rotating body.

When the stresses acting between the parts of a system depend only on the relative positions of those parts, the sum of the kinetic energy and potential energy of the system is always the same, provided the system be not acted upon by anything without it. Such a system is called conserva-tive, and is well illustrated by the swinging pendulum above referred to. But there are some stresses the direction of whose action depends on that of the relative motion of the visible bodies between which they appear to act, while there are others whose magnitude also depends on the relative velocities of the bodies. When work is done against these forces no equivalent of potential energy is produced, at least in the form in which we have been accustomed to recognize it, for if the motion of the system be reversed the forces will be also reversed and will still oppose the motion. It was long believed that work done against such forces was lost, and it was not till the present century that the energy thus transformed was traced, and the principle of conservation of energy established on a sound physical basis.

The principle of the Conservation of Energy has been stated by Professor Clerk Maxwell as follows :—

" The total energy of any body or system of bodies is a quantity which can neither be increased nor diminished by any mutual action of those bodies, though it may be trans-formed into any one of the forms of which energy is susceptible.

Hence it follows that, if a system be unaffected by any agent external to itself, the whole amount of energy pos-sessed by it will be constant, and independent of the mutual action of its parts. If work be done upon a system or energy communicated to it from without, the energy of the system will be increased by the equivalent of the work so done or by the energy so communicated; while if the sj'stem be allowed to do work upon external bodies or in any way to communicate energy to them, the energy of the system will be diminished by the equivalent of the work so done or energy so communicated.





In order to establish this principle it might at first sight appear necessary to make direct measurements of energy in all the forms in which it can possibly present itself. But there is one form of energy which can be readily measured, and to which all other forms can be easily reduced, viz., heat. If then we transform a quantity of energy from the form in which it is possessed by the earth and a raised weight, and which can be at once determined in foot-pounds or ergs, into heat, and measure the amount of heat so produced,—and if subsequently we allow an equal amount of energy to undergo various intermediate trans-formations, but to be finally reduced to heat,—and if we find that under all conditions the amount of heat is the same, and in different sets of experiments proportional only to the amount of energy with which we started, we shall be justified in asserting that no energy has been lost or gained during the transformations. It is the experimental proof of this which Joule has given us during the last thirty years, but we shall refer more at length to his work shortly.

It has been recently pointed out by Thomson and Tait {Natural Philosophy, arts. 262 sqq.) that Newton was acquainted with the principle of the conservation of energy, so far as it belongs purely to mechanics. But what became of the work done against friction and such non-conserva-tive forces was entirely unknown to Newton, and for long after his time this work was supposed to be lost. There were, however, some, even before Newton's time, who had more than a suspicion that heat was a form of energy. Bacon expressed his conviction that heat consists of a kind of motion or " brisk agitation " of the particles of matter. In the Novum Organum, after giving a long list of the sources of heat, some of which may fairly be adduced in support of his opinion, he says, " Erom these examples, taken collectively as well as singly, the nature whose limit is heat appears to be motion." In the following quotation Bacon appears to rise to the most complete appreciation of the dynamical nature of heat, nor do the most recent advances in science enable us to go much further. " It must not be thought that heat generates motion or motion heat (though in some respects this is true), but the very essence of heat, or the substantial self of heat, is motion and nothing else." Although Bacon's essay contains much sound reasoning, and many observations and experiments are cited which afford very strong evidence in favour of the theory he maintains, yet these are interspersed with so many false analogies, and such confusion between heat and the acrid or irritant properties of bodies, that we must re-serve for those who came after him the credit of having established the dynamical theory of heat upon a strictly scientific basis.

After Newton's time the first important step in the history of energy was made by Benjamin Thompson, Count Rumford, and was published in the Phil. Trans, for 1798. Rumford was engaged in superintending the boring of cannon in the military arsenal at Munich, and was struck by the amount of heat produced by the action of the boring bar upon the brass castings. In order to see whether the heat came out of the chips he compared the capacity for heat of the chips abraded by the boring bar with that of an equal quantity of the metal cut from the block by a fine saw, and obtained the same result in the two cases, from which he concluded that "the heat produced could not possibly have been furnished at the expense of the latent heat of the metallic chips."
Rumford then turned up a hollow cylinder which was cast in one piece with a brass six-pounder, and having reduced the connection between the cylinder and cannon to a narrow neck of metal, he caused a blunt borer to press against the hollow of the cylinder with a force equal to the weight of about 10,000 tt>, while the casting was made to rotate in a lathe. By this means the mean temperature of the brass was raised through about 70° Fahr., while the amount of metal abraded was only 837 grains. The cylinder, when it was subsequently removed from the rest of the casting, was found to weigh 113-13 lb.

In order to be sure that the heat was not due to the action of the air upon the newly exposed metallic surface, the cylinder and the end of the boring bar were immersed in 18-77 lb of water contained in an oak box. The temperature of the water at the commencement of the experiment was 60° Fahr., and after two horses had turned the lathe for 2 J hours the water boiled. Taking into account the heat absorbed by the box and the metal, Rumford calculated that the heat developed was sufficient to raise 26'58 lb of water from the freezing to the boiling point, and in this calculation the heat lost by radiation and conduction was neglected. Since one horse was capable of doing the work required, Rumford remarked that one horse can generate heat as rapidly as nine wax candles burning in the ordinary manner.

Finally, Rumford reviewed all the sources from which the heat might have been supposed to be derived, and concluded that it was simply produced by the friction, and that the supply was inexhaustible. " It is hardly necessary to add," he remarks, " that anything which any insulated body or system of bodies can continue to furnish without limitation cannot possibly be a material substance ; and it appears to me to be extremely difficult, if not quite impossible, to form any distinct idea of anything capable of being excited and communicated in the manner that heat was excited and communicated in these experiments, except it be motion."

About the same time that Rumford's experiments were published, Sir Humphry Davy showed that two pieces of ice could be melted by rubbing them together in a vacuum although everything surrounding them was at a temperature below the freezing point. He did not, however, see that since the heat could not have been supplied by the ice, for ice absorbs heat in melting, this experiment afforded conclusive proof of the dynamical nature of heat.

Though we may allow that the results obtained by Rumford and Davy demonstrate satisfactorily that heat is in some way due to motion, yet they do not tell us to what particular dynamical quantity heat corresponds. For example, does the heat generated by friction vary as the friction and the time during which it acts, or is it proportional to the friction and the distance through which the rubbing bodies are displaced,—that is, to the work done against friction,—or does it involve any other conditions'{ If it can be shown that, however the duration and all other conditions of the experiment maybe varied, the same amount of heat can in the end be always produced when the same amount of energy is expended, then, and only then, can we infer that heat is a form of energy, and that the energy consumed has been really transformed into heat. This Joule has done, and his experiments conclusively prove that heat and energy are of the same nature, and that all other forms of energy with which we are acquainted can be transformed into an equivalent amount of heat; and this is the condition ultimately assumed by the energy employed in doing work against friction and similar forces, which energy was in Newton's time supposed to be lost.

DEFINITION.—The quantity of energy which, if entirely converted into heat, is capable of raising the temperature of the unit mass of water from 0° C. to 1° C. is called the mechanical equivalent of heat.

One of the first who took in hand the determination of the mechanical equivalent of heat was Seguin, a nephew of Montgolfier. He argued that, if heat be energy, then, when it is employed in doing work, as in a steam-engine, some of the heat must itself be consumed in the operation. Hence he inferred that the amount of heat given up to the condenser of an engine when the engine is doing work must be less than when the same amount of steam is blown through the engine without doing any work. Seguin was unable to verify this experimentally, but in 1857 Him succeeded, not only in showing that such a difference exists, but in measuring it, and hence determining a tolerably approximate value of the mechanical equivalent of heat.

In 1839 S6guin endeavoured to determine the mechanical equivalent of heat from the loss of heat suffered by steam in expanding, assuming that the whole of the heat so lost was consumed in doing external work against the pressure to which the steam was exposed. This assumption, how-ever, cannot be justified, because it neglected to take account of work which might possibly have to be done within the steam itself during the expansion.

In 1842, Mayer, a physician at Heilbronn, published an attempt to determine the mechanical equivalent of heat from the heat produced when air is compressed. Mayer made an assumption the converse of that of Seguin, asserting that the whole of the work done in compressing the air was converted into heat, and neglecting the possibility of heat being consumed in doing work within the air itself or being produced by the transformation of internal potential energy. Joule afterwards proved (see below) that Mayer's assumption was in accordance with fact, so that his method was a sound one as far as experiment was concerned, and it was only on account of the values of the specific heats of air at constant pressure and at constant volume employed by him being very inexact that the value of the mechanical equivalent of heat obtained by Mayer was very far from the truth.

Passing over Colding, who in 1843 presented to the Royal Society of Copenhagen a paper entitled " Theses concerning Force," which clearly stated the " principle of the perpetuity of energy," and who also performed a series of experiments for the purpose of determining the heat developed by the compression of various bodies which entitle him to be mentioned among the founders of the modern theory of energy, we come to Dr Joule of Manchester, to whom we are indebted more than to any other for the establishment of the principle of the conservation of energy on the broad basis on which it now stands. The best known of Joule's experiments was that in which a brass paddle consisting of eight arms of complicated form arranged symmetrically round an axis was made to rotate in a cylindrical vessel of water containing four fixed vanes, which allowed the passage of the arms of the paddle but prevented the water from rotating as a whole. The paddle was driven by weights connected with it by strings which passed over friction rollers, and the temperature of the water was observed by thermometers which indicated ^jo-th of a degree Fahrenheit. Special experiments were made to determine the work done against resistances outside the ves-sel of water, which amounted to about '006 of the whole, and corrections were made for the loss of heat by radiation, the buoyancy of the air affecting the descending weights, and the energy dissipated when the weights struck the floor with a finite velocity. From these experiments Joule obtained 772'692 foot-pounds in the latitude of Manchester as equivalent to the amount of heat required to raise 1 lb of water through 1° Fahr. from the freezing-point. Adopt-ing the centigrade scale, this gives 1390846 foot-pounds as the mechanical equivalent of heat.

With an apparatus similar to the above, but smaller, made of iron and filled with mercury, Joule obtained results vary-ing from 772'814 foot-pounds when driving weights of about 58 lb. were employed to 775'352 foot-pounds when the driving weights were only about 19\ lb. By causing two conical surfaces of cast-iron immersed in mercury and contained in an iron vessel to rub against one another when pressed together by a lever, Joule obtained 776'045 foot-pounds for the mechanical equivalent of heat when the heavy weights were used, and 774'93 foot-pounds with the small driving weights. In this experiment a great noise was produced, corresponding to a loss of energy, and Joule endeavoured to determine the amount of energy necessary to produce an equal amount of sound from the string of a violoncello and to apply a corresponding correction.

The close agreement between the results of these experi-ments, differing widely as they do in their details, at least indicates that " the amount of heat produced by friction is proportional to the work done and independent of the nature of the rubbing surfaces." Joule inferred from them that the mechanical equivalent of heat is probably about 772 foot-pounds, or, employing the centigrade scale, about 1390 foot-pounds.

Previously to determining the mechanical equivalent of heat by the most accurate experimental method at his command, Joule established a series of cases in which the production of one kind of energy was accompanied by a disappearance of some other form. In 1840 he showed that when an electric current was produced by means of a dynamo-magneto-electric machine the heat generated in the conductor, when no external work was done by the current, was the same as if the energy employed in producing the current had been converted into heat by friction, thus show-ing that electric currents conform to the principle of the conservation of energy, since energy can neither be created nor destroyed by them. He also determined a roughly ap-proximate value for the mechanical equivalent of heat from the results of these experiments. Extending his investiga-tions to the currents produced by batteries, he found that the total voltaic heat generated in any circuit was pro-portional to the number of electrochemical equivalents electrolysed in each cell multiplied by the electromotive force of the battery. Now, we know that the number of electrochemical equivalents electrolysed is proportional to the whole amount of electricity which passed through the circuit, and the product of this by the electromotive force of the battery is the work done by the latter, so that in this case also Joule showed that the heat generated was proportional to the work done.

During his experiments on the heat produced by electric currents, Joule showed that, when a platinum wire was heated by the current so as to emit light, the heat generated in the circuit lor the same amount of work done by the battery was less than when the wire was kept cold, proving that when light is produced an equivalent amount of some other form of energy must disappear.

In 1844 and 1845 Joule published a series of researches on the compression and expansion of air. A metal vessel was placed in a calorimeter and air forced into it, the amount of energy expended in compressing the air being measured. Assuming that the whole of the energy was converted into heat, when the air was subjected to a pressure of 21'5 atmospheres Joule obtained for the mechanical equivalent of heat about 824-8 foot-pounds, and when a pressure of only 10 '5 atmospheres was employed the result was 796'9 foot-pounds.

In the next experiment the air was compressed as before, and then allowed to escape through a long lead tube im-mersed in the water of a calorimeter, and finally collected in a bell jar. The amount of heat absorbed by the air could thus be measured, while the work done by it in expanding could be readily calculated. In allowing the air to expand from a pressure of 21 atmospheres to that of 1 atmosphere the value of the mechanical equivalent of heat obtained was 821'89 foot-pounds. Between 10 atmospheres and 1 it was 815,875 foot-pounds, and between 23 and 14 atmospheres 76D74 foot-pounds.

But, unlike Mayer and S^guin, Joule was not content with assuming that when air is compressed or allowed to expand the heat generated or absorbed is the equivalent of the work done and of that only, no change being made in the internal energy of the air itself when the temperature is kept constant. To test this two vessels similar to that used in the last experiment were placed in the same calorimeter and connected by a tube with a stop-cock. One contained air at a pressure of 22 atmospheres, while the other was ex-hausted. On opening the stop-cock no work was done by the expanding air against external forces, since it expanded into a vacuum, and it was found that no heat was generated or absorbed. This showed that Mayer's assumption was true. On repeating the experiment when the two vessels were placed in different calorimeters, it was found that heat was absorbed by the vessel containing the compressed air, while an equal quantity of heat was produced in the calorimeter containing the exhausted vessel. The heat absorbed was consumed in giving motion to the issuing stream of air, and was reproduced by the impact of the particles on the sides of the exhausted vessel.1





The more recent researches of Dr Joule and Sir William Thomson (Phil. Trans., 1853, p. 357, 1854, p. 321, and 1862, p. 579) have shown that the statement that no internal work is done when a gas expands or contracts is not quite true, but the amount is very small in the cases of those gases which, like oxygen, hydrogen, and nitrogen, can only be liquefied by intense cold and pressure. It is worthy of note that mixtures of nitrogen and oxygen behaved more like theoretically perfect gases than either of the gases alone.

For the other contributions of Joule to our knowledge of energy, and for those of Sadi Carnot, Bankine, Clausius, Helmholtz, Sir William Thomson, James Thomson, Favre, and others, we must refer the reader to the articles on the several branches of physics, especially to HEAT.

Though we can convert the whole of the energy possessed by any mechanical system into heat, it is not in


Joule's papers will be found scattered through the Philosophical Magazine from 1839 to 1864 ; also in the Memoirs of the Manchester Society (2) vii. viii. ix. and (3) i. ; the Proceedings of the Manchester Society, 1859-60, 175 ; Phil. Trans., [1850] i. 61, [1853] 357, [1854] 321, [1859] 91, [1859] 133, [1863] 579 ; Proceedings of Roy. Soc., vi. 307, vi. 345, viii. 41, 178, viii. 355, viii. 556, viii. 564, ix. 3, ix. 254, ix. 496, x. 502 ; and the Reports of the British Asso-ciation [1859] ii. 12, and [1861] ii. 83.


our power to perform the inverse operation, and to utilize the whole of the heat in doing mechanical work. Thus we see that different forms of energy are not equally valuable for conversion into work. The energy of a system should be measured by the amount of work it can do under the most favourable conditions which can be imagined, though we are not necessarily capable of realizing them. The ratio of the portion of the energy of a system which can under given conditions be converted into work to the whole amount of energy present is called the availability of the energy. If a system be removed from all com-munication with anything outside of itself, the whole amount of energy possessed by it will remain the same, but will of its own accord tend to undergo such trans-formations as will diminish its availability: for since work is done only when energy undergoes transforma-tion, every change which it is allowed to undergo of its own accord deprives us of one opportunity of deriving useful work, that is, of converting a portion of the energy into the particular form we desire. This principle, known as the principle of the dissipation of energy, was first pointed out by Sir William Thomson in the Philosophical Magazine for April 1852, and was applied by him to some of the prin-cipal problems of cosmical physics. Though controlling all phenomena of which we have any experience, the principle of the dissipation of energy rests on a very different founda-tion from that of the conservation of energy; for while we can conceive of no means of circumventing the latter principle, it seems that the actions of intelligent beings are subject to the former only in consequence of the rudeness of the machinery which they have at their disposal for con-trolling the behaviour of those portions of matter in virtue of the relative motions or positions of which the energy with which they have to deal exists. If we have a weight capable of falling through a certain distance, we can employ the system consisting of the earth and weight to do an amount of useful work which is less than the potential energy possessed by the system only in consequence of the friction of the constraints, so that the limit of availability in this case is determined only by the friction which is unavoidable. Here we have to deal with a system with which we can grapple, and whose motions can be controlled at will. If, on the other hand, we have to deal with a system of molecules of whose motions we become conscious only by indirect means, while we know absolutely nothing either of the motions or positions of any individual molecules, it is obvious that we cannot grasp single molecules and control their movements so as to derive work from the system. All we can do, then, is to place the system under certain conditions, and be content with the amount of work which it is, as it were, willing to do under those conditions. It is well known that a greater pro-portion of the heat possessed by a body at a high tempera-ture can be converted into work than in the case of an equal quantity of heat possessed by a body at a low tempera-ture, so that the availability of heat increases with the temperature.

Clerk Maxwell supposed two compartments, A and B, to be filled with gas at the same temperature, and to be separated by a partition containing a number of trap-doors, each of which could be opened or closed without any expenditure of energy. An intelligent creature, or " demon," possessed of unlimited powers of vision, is placed in charge of each door, with instructions to open the door whenever a particle in A comes towards it with more than a certain velocity V, and to keep it closed against all particles in A moving with less than this velocity, but, on the other hand, to open the door whenever a particle in B approaches it with less than a certain velocity v, which is not greater than V, and to keep it closed against all particles in B moving with a greater velocity than this. By continuing this process every unit of mass which enters B will carry with it more energy than each unit which leaves B, and hence the temperature of the gas in B will be raised and that of the gas in A lowered, while no heat is lost and no energy expended, so that by the application of intelligence alone a portion of gas of uniform pres-sure and temperature may be divided into two parts, in which both the temperature and the pressure are different, and from which, therefore, work can be obtained at the expense of heat. If the gas do not liquefy, there seems no limit to the extent to which this operation may be carried, by increasing V and diminishing v, except that v cannot be made less than zero, which corresponds to the whole of the energy being abstracted from the gas in A and given to that in B. This shows that the principle of the dissipation of energy has control over the actions of those agents only whose faculties are too gross to enable them to grapple with those portions of matter in virtue of the relative motions or relative positions of which the energy exists with which they are concerned.

In April 1875 Lord Bayleigh published a paper in the Philosophical Magazine on " the work which may be gained during the mixing of gases." In the preface to the paper Lord Bayleigh says, " Whenever, then, two gases are allowed to mix without the performance of work, there is dissipation of energy, and an opportunity of doing work at the expense of low temperature heat has been for ever lost." He then shows that the amount of work obtainable is eqm.1 to that which can be done by the first gas in expanding into the space occupied by the second (supposed vacuous) together with that done by the second in expanding into the space occupied by the first. In the experiment imagined by Lord Bayleigh a porous diaphragm takes the place of the partition and trap-doors imagined by Clerk Maxwell, and the gases sort themselves on account of the difference in the velocities of mean square of molecules of the different gases. When the pressure on one side of the diaphragm is greater than that on the other, work may be done at the expense of heat in pushing the diaphragm, and the operation continued until the gases are uniformly diffused. There is this difference, however, between this experiment and Clerk Maxwell's, that when the gases have diffused the experiment cannot be repeated, and it is no more contrary to the dissipation of energy than is the fact that work may be derived at the expense of heat when a gas expands into a vacuum, for the working substance is not finally restored to its original condition; while Clerk Maxwell's experiment may be supposed to be continued and work obtained till the whole of the gas has been reduced to the absolute zero of temperature, and the ex-periment may be repeated by again heating the gas. Inde-pendently of Lord Bayleigh, Mr S. Tolver Preston, in November 1877, called attention to the work which may be done at the expense of heat during the diffusion of gases, and the bearing of this upon the dissipation of energy (see Nature, Nov. 8, 1877).

In these experiments the molecular energy of a gas is converted into work only in virtue of the molecules being separated into classes in which their velocities are different, and these classes then allowed to act upon one another through the intervention of. a suitable heat engine. If we could carry out this subdivision into classes as far as we pleased we might transform the whole of the heat of a body into work. The availability of heat is limited only by our power of bringing those particles whose motions constitute heat in bodies to rest relatively to one another; and we have precisely similar limits to the availability of the energy due to the motion of visible and tangible bodies.

If a battery of electromotive force E maintain a current C in a conductor, and no other electromotive force exist in the circuit, the whole of the work done will be converted into heat, and the amount of work done per second will be EC. If K. denote the resistance of the whole circuit, E = CR, and the heat generated per second is C2R. If the current drive an electromagnetic engine, the reaction of the engine will produce an electromotive force opposing the current. Suppose the current to be thus reduced to C. Then the work done by the battery per second will be EC or COR, while the heat generated per second will be C'2R, so that we have the difference (C - C')C'R for the energy consumed in driving the engine.

The ratio of this to the whole work 0 — C
done by the battery is —^—, so that the efficiency is

increased by diminishing C. If we could drive the engine so fast as to reduce C to zero, the whole of the energy of the battery would be available, no heat being produced in the wires, but the horse-power of the engine would be indefinitely small. The reason why the whole of the energy of the current is not available is that heat must always be generated in a wire in which a finite current is flowing, so that, in the case of a battery in which the whole of the energy of chemical affinity is employed in producing a current, the availability of the energy is limited only on account of the resistance of the conductors, and may be increased by diminishing this resistance. The availability of the energy of electrical separation in a charged Leyden jar is also limited only by the resistance of conductors, in virtue of which an amount of heat is necessarily produced, which is greater the less the time occupied in discharging the jar. The availability of the energy of magnetization is limited by the coercive force of the magnetized material, in virtue of which any change in the intensity of magnetization is accompanied by the production of heat.

Since the motion of the centre of mass of a system is unaffected by any actions taking place between the parts of the system, it is plain that a system considered by itself cannot be said to possess energy in virtue of the motion of its centre of mass, and in estimating the energy of the system at any instant we may therefore treat this point as fixed, and consider only motions relative to it. Thus any motion of rotation we may consider to take place about an axis through the centre of mass. Now, if a system be not acted upon by any forces from without which have a moment about this axis, the product of the angular velocity of the system and of its moment of inertia about the axis of rotation will remain unchanged. Hence if we increase the moment of inertia we shall diminish the angular velocity in the inverse ratio, and therefore diminish the energy of rotation in this ratio, since the latter is propor-tional to the moment of inertia and the square of the angular velocity. If, then, we have a material system moving in the most general manner possible, we shall -educe its kinetic energy to a minimum by causing such ictions to take place between the parts of the system as will make its moment of inertia about the invariable line as great as possible, and then changing the relative motions of the parts in such a manner that they move as if they were rigidly connected with one another. The motion of the system, will then be a simple rotation with its kinetic energy as small as possible, and the greatest amount of energy will thus have been transformed.

In all the cases we have examined there is a general tendency for other forms of energy to be transformed into heat on account of the friction of rough surfaces, the resistance of conductors, or similar causes, and thus to lose availability. In some cases, as when heat is converted into the kinetic energy of moving machinery or the potential energy of raised weights, there seems to be an ascent of energy from the less available form of heat to the more available form of mechanical energy, but when this takes place there is always, accompanying it, a quantity of heat which passes from a body at a high temperature to one at a lower temperature, thus losing availability, so that on the whole there is a degradation of energy. Thus Thomson's second law of thermodynamics, which states that " it is impossible by means of inanimate material agency to obtain work by cooling matter below the temperature of the coldest body in the neighbourhood," appears to be generally true, except when this work is obtained at the expense of some other condition of advantage, as, for example, that possessed by air at a higher pressure than the surrounding atmosphere, or by different kinds of matter which are separate and tend to diffuse, and then the work having once been obtained, the system cannot be restored to its original condition without the degradation of energy from some other source, even though the heat converted into work be restored to the working bodies.

It is sometimes important to consider the rate at which energy may be transformed into useful work, or the horse-power of the agent. It generally happens that to obtain the greatest possible amount of work from a given supply of energy, and to obtain it at the greatest rate, are conflicting interests. We have seen that the efficiency of an electromagnetic engine is greatest when the current is indefinitely small, and then the rate at which it works is also indefinitely small. Jacobi showed that for a given electromotive force in the battery the horse-power is greatest when the current is reduced to one-half of what it would be if the engine were at rest. A similar condition obtains in the steam-engine, in which a great rate of working necessitates the dissipation of a large amount of energy through the resistance of the steam-pipes, etc. The only way to secure a high degree of efficiency with a great horse-power in the case of the steam-engine is by increasing the section of the steam-pipes and the areas of the steam ports. The efficiency of an electromagnetic engine cannot be greater than one-half when it is working at its maximum horse-power, but we may obtain any fixed rate of working we please with a given degree of efficiency by diminishing the resistance of the battery and conductors until the maximum horse-power of the engine exceeds that at which it is to be worked by a sufficient amount. (W. G.)