Thus we see that finally we should be led to conclude that the equations which define distances are of an order superior to the second. Why should we be shocked at that, why do we find it perfectly natural for the series of phenomena to depend upon the initial values of the first derivatives of these distances, while we hesitate to admit that they may depend on the initial values of the second derivatives? This can only be because of the habits of mind created in us by the constant study of the generalized principle of inertia and its consequences.

The values of the distances at any instant depend upon their initial values, upon those of their first derivatives and also upon something else. What is this _something else_?

If we will not admit that this may be simply one of the second derivatives, we have only the choice of hypotheses. Either it may be supposed, as is ordinarily done, that this something else is the absolute orientation of the universe in s.p.a.ce, or the rapidity with which this orientation varies; and this supposition may be correct; it is certainly the most convenient solution for geometry; it is not the most satisfactory for the philosopher, because this orientation does not exist.

Or it may be supposed that this something else is the position or the velocity of some invisible body; this has been done by certain persons who have even called it the body alpha, although we are doomed never to know anything of this body but its name. This is an artifice entirely a.n.a.logous to that of which I spoke at the end of the paragraph devoted to my reflections on the principle of inertia.

But, after all, the difficulty is artificial. Provided the future indications of our instruments can depend only on the indications they have given us or would have given us formerly, this is all that is necessary. Now as to this we may rest easy.

CHAPTER VIII

ENERGY AND THERMODYNAMICS

ENERGETICS.--The difficulties inherent in the cla.s.sic mechanics have led certain minds to prefer a new system they call _energetics_.

Energetics took its rise as an outcome of the discovery of the principle of the conservation of energy. Helmholtz gave it its final form.

It begins by defining two quant.i.ties which play the fundamental role in this theory. They are _kinetic energy_, or _vis viva_, and _potential energy_.

All the changes which bodies in nature can undergo are regulated by two experimental laws:

1 The sum of kinetic energy and potential energy is constant. This is the principle of the conservation of energy.

2 If a system of bodies is at _A_ at the time t_{0} and at _B_ at the time t_{1}, it always goes from the first situation to the second in such a way that the _mean_ value of the difference between the two sorts of energy, in the interval of time which separates the two epochs t_{0} and t_{1}, may be as small as possible.

This is Hamilton's principle, which is one of the forms of the principle of least action.

The energetic theory has the following advantages over the cla.s.sic theory:

1 It is less incomplete; that is to say, Hamilton's principle and that of the conservation of energy teach us more than the fundamental principles of the cla.s.sic theory, and exclude certain motions not realized in nature and which would be compatible with the cla.s.sic theory:

2 It saves us the hypothesis of atoms, which it was almost impossible to avoid with the cla.s.sic theory.

But it raises in its turn new difficulties:

The definitions of the two sorts of energy would raise difficulties almost as great as those of force and ma.s.s in the first system. Yet they may be gotten over more easily, at least in the simplest cases.

Suppose an isolated system formed of a certain number of material points; suppose these points subjected to forces depending only on their relative position and their mutual distances, and independent of their velocities. In virtue of the principle of the conservation of energy, a function of forces must exist.

In this simple case the enunciation of the principle of the conservation of energy is of extreme simplicity. A certain quant.i.ty, accessible to experiment, must remain constant. This quant.i.ty is the sum of two terms; the first depends only on the position of the material points and is independent of their velocities; the second is proportional to the square of these velocities. This resolution can take place only in a single way.

The first of these terms, which I shall call _U_, will be the potential energy; the second, which I shall call _T_, will be the kinetic energy.

It is true that if _T_ + _U_ is a constant, so is any function of _T_ + _U_,

{Phi}(_T_ + _U_).

But this function {Phi}(_T_ + _U_) will not be the sum of two terms the one independent of the velocities, the other proportional to the square of these velocities. Among the functions which remain constant there is only one which enjoys this property, that is _T_ + _U_ (or a linear function of _T_ + _U_, which comes to the same thing, since this linear function may always be reduced to _T_ + _U_ by change of unit and of origin). This then is what we shall call energy; the first term we shall call potential energy and the second kinetic energy. The definition of the two sorts of energy can therefore be carried through without any ambiguity.

It is the same with the definition of the ma.s.ses. Kinetic energy, or _vis viva_, is expressed very simply by the aid of the ma.s.ses and the relative velocities of all the material points with reference to one of them. These relative velocities are accessible to observation, and, when we know the expression of the kinetic energy as function of these relative velocities, the coefficients of this expression will give us the ma.s.ses.

Thus, in this simple case, the fundamental ideas may be defined without difficulty. But the difficulties reappear in the more complicated cases and, for instance, if the forces, in lieu of depending only on the distances, depend also on the velocities. For example, Weber supposes the mutual action of two electric molecules to depend not only on their distance, but on their velocity and their acceleration. If material points should attract each other according to an a.n.a.logous law, _U_ would depend on the velocity, and might contain a term proportional to the square of the velocity.

Among the terms proportional to the squares of the velocities, how distinguish those which come from _T_ or from _U_? Consequently, how distinguish the two parts of energy?

But still more; how define energy itself? We no longer have any reason to take as definition _T_ + _U_ rather than any other function of _T_ + _U_, when the property which characterized _T_ + _U_ has disappeared, that, namely, of being the sum of two terms of a particular form.

But this is not all; it is necessary to take account, not only of mechanical energy properly so called, but of the other forms of energy, heat, chemical energy, electric energy, etc. The principle of the conservation of energy should be written:

_T_ + _U_ + _Q_ = const.

where _T_ would represent the sensible kinetic energy, _U_ the potential energy of position, depending only on the position of the bodies, _Q_ the internal molecular energy, under the thermal, chemic or electric form.

All would go well if these three terms were absolutely distinct, if _T_ were proportional to the square of the velocities, _U_ independent of these velocities and of the state of the bodies, _Q_ independent of the velocities and of the positions of the bodies and dependent only on their internal state.

The expression for the energy could be resolved only in one single way into three terms of this form.

But this is not the case; consider electrified bodies; the electrostatic energy due to their mutual action will evidently depend upon their charge, that is to say, on their state; but it will equally depend upon their position. If these bodies are in motion, they will act one upon another electrodynamically and the electrodynamic energy will depend not only upon their state and their position, but upon their velocities.

We therefore no longer have any means of making the separation of the terms which should make part of _T_, of _U_ and of _Q_, and of separating the three parts of energy.

If (_T_ + _U_ + _Q_) is constant so is any function [phi](_T_ + _U_ + _Q_).

If _T_ + _U_ + _Q_ were of the particular form I have above considered, no ambiguity would result; among the functions [phi](_T_ + _U_ + _Q_) which remain constant, there would only be one of this particular form, and that I should convene to call energy.

But as I have said, this is not rigorously the case; among the functions which remain constant, there is none which can be put rigorously under this particular form; hence, how choose among them the one which should be called energy? We no longer have anything to guide us in our choice.

There only remains for us one enunciation of the principle of the conservation of energy: _There is something which remains constant_.

Under this form it is in its turn out of the reach of experiment and reduces to a sort of tautology. It is clear that if the world is governed by laws, there will be quant.i.ties which will remain constant.

Like Newton's laws, and, for an a.n.a.logous reason, the principle of the conservation of energy, founded on experiment, could no longer be invalidated by it.

This discussion shows that in pa.s.sing from the cla.s.sic to the energetic system progress has been made; but at the same time it shows this progress is insufficient.

Another objection seems to me still more grave: the principle of least action is applicable to reversible phenomena; but it is not at all satisfactory in so far as irreversible phenomena are concerned; the attempt by Helmholtz to extend it to this kind of phenomena did not succeed and could not succeed; in this regard everything remains to be done. The very statement of the principle of least action has something about it repugnant to the mind. To go from one point to another, a material molecule, acted upon by no force, but required to move on a surface, will take the geodesic line, that is to say, the shortest path.

This molecule seems to know the point whither it is to go, to foresee the time it would take to reach it by such and such a route, and then to choose the most suitable path. The statement presents the molecule to us, so to speak, as a living and free being. Clearly it would be better to replace it by an enunciation less objectionable, and where, as the philosophers would say, final causes would not seem to be subst.i.tuted for efficient causes.

THERMODYNAMICS.[4]--The role of the two fundamental principles of thermodynamics in all branches of natural philosophy becomes daily more important. Abandoning the ambitious theories of forty years ago, which were enc.u.mbered by molecular hypotheses, we are trying to-day to erect upon thermodynamics alone the entire edifice of mathematical physics.

Will the two principles of Mayer and of Clausius a.s.sure to it foundations solid enough for it to last some time? No one doubts it; but whence comes this confidence?

[4] The following lines are a partial reproduction of the preface of my book _Thermodynamique_.

An eminent physicist said to me one day _a propos_ of the law of errors: "All the world believes it firmly, because the mathematicians imagine that it is a fact of observation, and the observers that it is a theorem of mathematics." It was long so for the principle of the conservation of energy. It is no longer so to-day; no one is ignorant that this is an experimental fact.