What distinguishes the stone moving like a pendulum from the stone which simply falls is, that at its lowest point it has not remained lying still, but possesses a certain velocity. By means of this it lifts itself again, and after it has reached its former height, it has lost its velocity. Therefore, there is a reciprocal relation between the work which it loses and the velocity which it gains, and the question may therefore be put, How can this relation be represented mathematically? Experience teaches that in every such case a function of the velocity and of another property of the body, called mass, can be established in such a way that this function, called the kinetic energy of the body, increases precisely as much as the amount of work the body has expended, and vice versa. The sum of the kinetic energy of the body and of the work is therefore constant, and the clearest mode of conceiving of this relation is by assuming that work can be transformed into kinetic energy and vice versa in such a way that given amounts of the two magnitudes are equal or equivalent to one another. Naturally, this is only an abbreviated way of expressing the actual relations, for it might just as well be assumed that the work really disappears and the kinetic energy really originates anew, and that the disappearance of the one substance only happens regularly to coincide with the origin of the other. But it is this regular conjunction of phenomena that constitutes the sole ground of every causal relation, and in such a sense we are justified in regarding the disappearing work as the cause of the kinetic energy that arises, and to designate this relation summarily as a transformation.
By the inclusion of cases in which work is converted into kinetic energy the law of the conservation of work therefore becomes the law of the conservation of the sum of work and kinetic energy. We are thereby compelled to extend the concept of substance, which at first contains only work, to the sum of both magnitudes, and to introduce a new name for this enlarged concept.
It will soon appear that all cases of imperfect machines, in which work disappears without giving rise to an equivalent amount of kinetic energy, can, with a corresponding enlargement of the concept, be likewise included in the law of conservation. For experience has shown that in such cases something else arises, heat, light, or electric force, etc. This generalized concept, which embraces all natural processes and permits the sum of all corresponding values to be expressed by a law of conservation, we call energy. The law in question, therefore, is:
In all processes the sum of the existing energies remains unchanged.
The principle of the conservation of work in perfect machines proves to be an ideal special instance of this general law. A perfect machine is one in which work changes into nothing but work of another kind, and not into a different kind of energy. Then each side of the equation which expresses the general law of energy, namely,
Energy that has disappeared = energy that has arisen,
contains only the magnitude of the work, and expresses the law of the conservation of work. If, on the other hand, as in the case of the pendulum, the work increasingly changes part by part into kinetic energy, and vice versa, the equation during the first period is:
Work that has disappeared = kinetic energy that has arisen,
and during the second period in which the pendulum rises again,