the atoms which separate and combine again must at first move confusedly about in irregular disorder till the new regular arrangement is found. A still more striking example of a regular law arising from chance events is afforded by meteorological phenomena, the varied change of which follows a law that is clearly recognisable from the means of long periods.

In all these cases, and in our theory as well, the regularity arises only from the great number of the elementary processes from which it results. If this number were not so great, the result in similar cases would not always be absolutely and fixedly one and the same, but there would be different results conceivable of more or less probability. But the greater this number, the greater the probability that of all possible consequences a single perfectly definite one would occur just as if it were directly caused by the operation of a fixed law of nature instead of by the play of numberless casual events.

Applied to our theory this general view teaches that in a really infinite number of molecules of gas a condition must exist, the law of which must admit of recognition and even of mathematical expression, in spite of the chance character of the motion of each individual molecule. We must therefore be able to determine how many molecules per thousand, say, taken at random from the countless swarm, have a speed of a definite magnitude, or, in somewhat different words, we must be able to express numerically the chance that any given molecule in a region filled with an infinite host of molecules should attain a speed of given value.

In a gaseous mass consisting of a finite number of molecules this condition of simple regularity will not be attained, as in the case of an infinite number-at least not at every instant. In a finite system this law shows itself only if all the states which occur with continual change in the course of longish periods are taken into account together. The particular condition of regularity which is exhibited at every moment by an infinite swarm determines also for a finite number of molecules its mean state during a considerable period.

The arrangement that changes from moment to moment and represents the distribution of the different speeds among a limited number of molecules, oscillates therefore about a mean regular state, and in such wise that the result which ensues in the course of a considerable time is the same as if that regular state had existed at each moment. In the calculation of the result we may therefore consider that regular distribution of speeds as always existing instead of the actual circumstances of constant change.

24. Maxwell's Law

The law which regulates this distribution of speeds among the gaseous molecules was discovered by James Clerk Maxwell, who thus made it possible to calculate, by strict mathematical methods, the mean values of the speeds which hitherto had been only estimated, and a knowledge of which was necessary for the development of the theory of gases.

Maxwell's law of distribution, the theoretical foundation of which rests on the calculus of probabilities, agrees exactly in form with another law which is also founded on this calculus. The possible values which the components of the molecular velocities can assume are distributed among the molecules in question according to the same law as the possible errors of observation are by the method of least squares distributed among the observations.

According to this law the equilibrium of a gas depends, of course, as was to be expected, in no way upon equality of motion in all the particles. All values between 0 and ∞ occur for the components of velocity, and in such fashion that small values occur oftener than large ones, just as according to the method of least squares errors of small magnitude should happen oftener than large ones.

In order to give an idea of this law without having recourse to mathematical formulæ I will quote a few figures.2 1 Phil. Mag. [4] xix. 1860, p. 22; Scientific Papers, i. p. 377.

2 Obtained from the values of the integrals

=0.00413.

S'dz e-** = 0·74682, ["dz e ̄*=0·13525, ["dz e-*=0

If 10,000 molecules move with such velocities that their components in any given direction lie in magnitude between O and a certain value W (see § 27), then there are only 1,811 for which this component lies between W and 2W, and but fifty-five with a value between 2W and 3W for the component. The small values therefore predominate in remarkably large proportion, and the probability of larger values of a component of the molecular velocity, just as that of large errors of observation, is vanishingly small.

In this form the law expresses the frequency of occurrence of the values which the three components of the velocity assume. We shall show later on, in § 26, how the probability of a particular value of the resultant velocity of a molecule can be deduced from it.

As has been already mentioned, Maxwell's law can be employed in two ways. First of all it tells us how many of a certain number of molecules move with a given velocity at the same moment; but, secondly, it serves equally well to give the frequency with which one and the same particle attains a given velocity in consequence of its encounters with other particles.

25. Proof of Maxwell's Law

Several demonstrations resting on different footings have been tentatively given for this law of distribution of molecular speeds.

Its discoverer, J. Cl. Maxwell, first1 proved it by the assumption of a principle which, though true, itself needs proof. Since Maxwell himself recognised this defect, he later gave a second proof,3 the basis of which is subject to no doubt. Since the state of equilibrium with which the law is concerned is not disturbed by encounters between the molecules, but is continuously maintained, every change produced by collision must at once be cancelled by other collisions. A velocity of a particular magnitude and direction

1 Phil. Mag. [4] xix. 1860, p. 22; Scientific Papers, i. p. 377.

2 See the end of § 14* of the Mathematical Appendices.

Phil. Trans. clvii. 1867, p. 49; Phil. Mag. [4] xxxv. 1868, p. 185; Scientific Papers, ii. p. 43.

will therefore result from one collision as often as it will be destroyed by another; and from this principle, together with the laws of collision, Maxwell's law may be established.

L. Boltzmann1 completed and perfected this proof by employing stricter mathematical work, and thus removing just ground for doubt. A further step forward we owe to H. A. Lorentz, who raised a new objection and improved the calculation, thereby inciting Boltzmann3 to again give a new proof, which proof may now be considered as quite free from objection.

Further, Kirchhoff has given a proof of the law in his Lectures; but against this, too, according to a remark of Boltzmann, objection may be made.

In a different way the proof of this law was attempted in the first edition of this book. The weak points of this attempt were removed by N. N. Pirogoff, and a varied form of Pirogoff's proof is given in the second of the Mathematical Appendices.

These mathematical proofs cannot be repeated here, nor should we attempt here to give them; I will only indicate a striking point that arises from them.

Since the law of distribution which we are looking for is concerned with the state which in time results from the encounters between molecules, we might expect that a knowledge of what occurs during the encounters might be necessary in order to find the law. It would seem that we ought to know the law of collision for molecules if we would calculate the final result of the collisions; and apparently we must therefore know whether the molecules behave during collision as elastic bodies, or whether their collisions occur as those between hard or soft bodies."

Wiener Sitzungsber. Iviii. 1868, p. 517; lxvi. 1872, p. 275.

2 Ibid. xcv. 1887, p. 115.

Ibid. xov. 1887, p. 153.

• Vorlesungen über mathematische Physik, iv. (Theorie der Wärme, herausgeg. von Planck), 1894, p. 142 (14th Lecture).

Münchener Sitzungsber. xxiv. 1894, p. 207. Ibid. (Planck) p. 391.

Journal der russ. physik.-chem. Ges. xvii. 1885, pp. 114-135, 281-313. A discussion of the question how far the laws of elastic collision are applicable to molecules of gas is given in Chapter X.

But the knowledge of the laws of collision proves to be quite unnecessary for the proof of Maxwell's law. So little indeed is it necessary, that Maxwell was able in course of time to change his views on this point without having to upset or reject the theory established by him. At first he thought it probable that two colliding molecules, just as two hard elastic bodies, would, after the collision in which they might have come into actual contact, be hurled apart by a suddenly arising and vanishing force. But later, on grounds which I shall examine in Chapter X. § 123, in discussing the molecular forces, he declared it more probable that two molecules of gas act on each other with repulsive forces, which, varying inversely as the fifth power of the distance, are insensible at greater distances, but at smaller suffice to force apart two molecules coming very near each other. We may hold either the one view or the other without prejudicing the validity of these proofs.

This shows that for the purpose we have first in view it would be superfluous to indulge in speculations on the character and the laws of the forces that come into play during the collision of molecules. A decision in favour of a particular hypothesis would only diminish the value of Maxwell's law, as it would seem to limit its validity. For the law is valid independently of all hypotheses.

For the proof of Maxwell's law it is therefore sufficient, as Boltzmann' has already recognised, to impress the quite general, and on that account indubitable, propositions of analytical mechanics; Maxwell too has made use of these alone. So in the proof given in the Mathematical Appendices (§ 10*) it stands out clearly that Maxwell's law needs the assumption of these general propositions only.

The most important of these propositions is that of the Conservation of Energy. The admissibility of its application to molecular motion will not be questioned, but there is one precaution to mind. If the molecules do not consist of single massive points, but are made up of combinations of several atoms, we have to take into account not merely 1 Wiener Sitzungsber. lxiii. 1871, pp. 397, &c.