APPENDIX II

MAXWELL'S LAW

9. On Some Older Proofs

THE state of equilibrium of a multitude of molecules of gas, as has been already shown in § 22, does not consist in their all moving with equal speed. On the contrary, the velocity of any particle changes at every encounter, not only in direction but also in magnitude. But the values of the speed fluctuate about a mean value. The law of deviation of the actual speeds of the particles from this mean value was first perceived by Maxwell; he found that the components of the molecular velocity are distributed among the particles of a gas in equilibrium with the same regularity as we find in all apparently fortuitous phenomena and processes which are really subject to fixed changes. For the distribution of the speeds the same law holds good which, according to Gauss, regulates the distribution of chance errors of observation among the several observations.

Very many proofs have been given of Maxwell's law. One such proof was attempted in the first edition of this book, wherein the law was put forward as the most probable of all conceivable laws. Although the mathematical investigation of this idea is closely connected with the proof given by Gauss' of the method of least squares, the proof in its first form cannot be admitted as valid, and the doubts thrown upon it by Boltzmann2 and von Kries must be held to be well founded.

N. N. Pirogoff, however, showed that my proof can be

p. 192.

Theoria motus corp. cæl. §§ 175-177.

Wiener Sitzungsber. lxxvi. 1877, p. 373.

Principien der Wahrscheinlichkeitsrechnung, Freiburg 1886, Chap. VIII.

Journ. d. Russ. phys.-chem. Ges. xvii. 1885, pp. 114-135, 281-313. Abstracted in Fortschr. d. Physik, 1886, pt. 2, p. 237. Exner's Repertorium, xxvii. 1891, p. 540.

justified if the mathematical formulæ are differently interpreted. The differential calculus expresses the property of a function in having a maximum in the same way as its behaviour in maintaining its value when its argument is varied. My formula, which I again give in § 12* in unchanged form, need not therefore contain the meaning that Maxwell's law is the most probable of all conceivable laws; but they show, as Boltzmann had already recognised before Pirogoff, that among a limited number of molecules the values of the speeds may be distributed in different ways, and that all these different ways possess an equal degree of probability. On this theorem Pirogoff founded his proof, which he carried out by the same process as I did mine. Since this altered proof by Pirogoff was originally published in Russian only, and is therefore little known elsewhere, I will here give his method at length.

Pirogoff starts with the assumption that out of an unlimited number of gaseous molecules, whose motions have already become in accordance with Maxwell's law, a group of N particles is so picked out that the choice is guided only by chance. He then investigates the probability that given values of the velocitycomponents u, v, w are to be found in this group. By these values also the average state of the motion of the group, its average speed and energy, are determined. If now a second group of N particles is again picked out by chance, there will be other values of the components in this second group; but the average values of the speed and energy may, in spite of this, be the same as with the first group. The probability that each group will have the same average value is the same for both.

That we may arrive at the formulæ of my former proof by the stricter way suggested by Pirogoff was shown me on February 5, 1882, by Gustav Lübeck, with whom I was then corresponding on the subject of my memoir and his. I had then, unfortunately, no opportunity of making use of this communication. I will therefore now lay the foundation of this proof in a way which will, I hope, be valid as a more comprehensive improvement.

1 Festschrift zur zweiten Säcularfeier des Friedrichswerderschen Gymnasiums in Berlin, 1881, p. 295; Ueber die Bewegung eines kugelförmigen Atoms.

B B

10*. Hypotheses Used in the Proof

Maxwell's law of distribution refers to the state which a group of gaseous molecules finally attains as its state of equilibrium in consequence of their encounters. If this state once occurs it is maintained to the last in unchanged fashion. But, strictly speaking, it can only be reached when the number of gaseous molecules is unlimited; for by an encounter any value whatever of velocity may result, and only with an infinite number of particles can all possible values of the speed be actually existing at each moment.

If the number of particles is limited, Maxwell's law must be understood otherwise. Since the state of motion of the group of particles is altered in a perceptible degree at every single encounter between two particles, Maxwell's distribution cannot exist at every moment, but will occur with exactness only when the changing states which succeed each other in the course of a sufficiently long period are all taken into account together. If all these different distributions did not succeed each other, but occurred simultaneously together in an unlimited number of particles, the law would not thereby be changed; but Maxwell's law must be equally valid in both cases.

After this remark we can proceed to investigate more closely the function required which expresses the value of the probability. For this purpose let us consider a large number Z, say 1,000 or 100,000, of the changing states succeeding each other, which a group of N particles pass through. On the whole, then, NZ different states of a single particle come into account. Among these numerous cases it will often happen that a given particle m, attains a velocity the components of which in three rectangular directions are u1, v1, w1. The number of cases in which this occurs we may represent by a function of the form

NZF(u, v, w);

for it must be proportional to the number NZ of states, and it must further depend on the values u, v, w, of the components. For Z∞ the value of the function

F1 = F(u, v1, w1)

which expresses this law of dependence is the probability of

occurrence of the possible case that the particle m, should move with a velocity made up of the components u1, V1, W1.

Although the motion of a particle is not independent of the motions of the other N 1 particles, yet the function F, will be determined only by the three arguments u, v, w, if the NZF, cases are so counted that account is taken in them only of the state of the one particle m1, and not of the states of the other particles also.

For the same reason, if the particies of the group are all like each other, the probability of the event that a second particle ma has the components u2, v2, w, is determined by the same function with different arguments, viz.

From these two values of the probability-function we easily obtain by a known law the expression for the probability of the occurrence of both sets of circumstances, viz. that the particle m1 should have the components u, v1, w1, and m, the components U2, V2, W2. It is necessary to assume only that the number Z, which we look upon as very large, may be approximately taken as infinitely large, and that the function F is determined in correspondence with this assumption; in this case the two events of my possessing the components u, v, w, and of m, having U2, V2, w2 are independent of each other, and the probability of their simultaneous occurrence is therefore expressed by the product of the two functions, and therefore by 1

F1F2 = F(u1, v1, w1)F(u2, v2, W2).

2

If we also consider a third particle m, which may have the velocities 3, 3, w3, a fourth with components 4, V4, w4, and so on for all the N particles which form the group, we have in the product of the N factors

FF... FF(u1, v1, w1)F(u2, V2, W2)... F(Ux, VN, WN)

This formula and those which follow later would contain numerical factors which would have to be formed according to the rules of combinations if we did not fix a definite series of the particles. If we sought the probability that one of two particles m, and m, had the components uj, v1, w1, and the other the components u, v, w, this would be twice as great as in the case we have taken. Since these factors have no influence on the result, it would be superfluous to complicate the formulæ by inserting them. The factors, furthermore, disappear when the number of particles considered approxi mates to infinity. (Encke, Astron. Jahrb. für 1834, p. 256.)

the chance of the event that, among all the changing states of the group, the first particle m, has the components u1, vi, w1, the second my the components aug, 02, 202, and so on, the last my having the components UN, VN, WN.

In this nothing is assumed regarding the time at which these values of the velocity for the individual particles of the group occur. We may therefore apply the formula to the values of the components which the N particles may have at any given moment whatever.

At another time the particles have different velocities, and u', v', w' may then be the components of m1, also u'2, v'2, W': those of mg, &c. The probability of this changed state is then given by the product

F'F'2... F'N = F(u'1, v'1, w',)F(u′2, v′2, w′2) . . . F'(u'x, v′N, W'x),

which contains the same function F as the first, but with different arguments.

The two products are equal to each other in value, for each of the states of distribution is as likely as the other, because, according to our supposition, both form part of the state of equilibrium which finally ensues. For equilibrium, therefore, it results that the function F must satisfy the equation

FF...FFF'2... FN,

or that the product

F(u1, v1, w¡)F(u2, V2, W2). . . F(ux, VN, Wx)

must always have one and the same value for all systems of the values of the variables that occur.1

This theorem is proved differently by Pirogoff (Journ. d. russ. phys. chem. Ges. 1885, xvii.). Pirogoff considers an infinite number of gaseous particles which are in a state of equilibrium. From this infinite multitude N particles are taken out. The probability of finding given values of the components u, v, w among these N particles is expressed by the given product. Of the same magnitude is the chance of taking a second group of N other particles which, though having different components of velocity from those of the first group, have the same total kinetic energy and the same motion of their centroid. Maxwell's law follows likewise from this assumption.