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.” 1he most useful unit for scientific purposes is one which depends only on the fundamental units of length, mass, aud 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 required 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 a gramme, and the erg is the amount of work required to raise τ ⅜ τ °f 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 iβ 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 position. 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 of — centimetres, where a represents the numerical value

2y 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 ∕^2 * β represented by mg^r, that is, ∣wzv 2 ergs. But it is merely 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 projection it possessed imv 2 units of energy. Now, whatever be the direction in which a body is moving, a frictionless constraint, like a string attached to the body, can cause its velocity to be changed into the vertical direction 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, ⅛rrιv i 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 therefore possessed in common by the system consisting of the earth and the body; and the energy which a system possesses 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 pendulum. When at the limits of its swing, the pendulum is for au 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 ⅛mv 2 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 ν relative to the other, we can only obtain ⅛mv 2 units of work from this system alone, and we ought not to say that the system considered by itself possesses more than ⅛mν 2 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 ^mv 2 , 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 ⅛mv 2 will not be a true representation 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 ∣S(mυ 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 restis 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 ω, the work which can be done by it if we can succeed in bringing it to rest will be ∣Iω 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 conservative, 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 transformed 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 possessed 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 system 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 footpounds 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 transformations, 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-conservative 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, “From 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 reserve 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 lb, 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 113Ί3 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∣ 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 zvithout 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...