the energy of the molecular motion, but also that of the motions of the individual atoms, which presumably consist of oscillations and rotations about the common centroid.

The proposition of the conservation of energy makes it necessary to admit certain assumptions, which, indeed, would be contested by no one, but which are only hypotheses, since they fail to possess the certainty afforded by a mathematical proof. In the particular case when we apply this proposition to a system of material particles endowed with mutual attraction or repulsion, the most essential of these hypotheses is that the action of one particle on another is equal to the reaction exerted on the first by the second.

If this assumption is admitted, no further hypothesis is needed for the proof of Maxwell's law save the proposition of the conservation of energy, the other general theorems of analytical mechanics which are used for the mathematical proof being immediate consequences of this assumption regarding action and reaction. This applies especially to the theorem of the Conservation of Momentum of the centroid of a system-a theorem which is brought into our proof only when the gas possesses a general forward velocity in addition to its internal heat-motion, i.e. when it is in a state of flow. The same is true with regard to the socalled theorem of the Conservation of Areas or of the Moments of Momenta; this comes into consideration only when the gas is in a state of rotation.

If then no further hypotheses are needed for the proof of Maxwell's law, the assumption underlying the first proofs, viz. that the molecules are simple material points, can especially be dispensed with. The law therefore holds not only for monatomic molecules, i.e. such as consist of a single atom or one massive point, but also for polyatomic molecules or chemical combinations of several atoms.

That this extension of Maxwell's law to compound molecules is admissible was first recognised by Boltzmann' who has thereby exercised a very important influence on the further development of the theory.

It must, however, not be overlooked that the law has

Wiener Sitzungsber. lxiii. 2. Abth. 1871, p. 397.

reference only to the molecular motion in which all the components of the molecule equally take part. For the atomic motions that exist by its side, the motions, that is, which the atoms separately possess within the molecule, the law can hold only with a certain limitation, since there is a difference, which we shall discuss in §§ 57 and 60, between the free motion of a molecule and the constrained motion of the atoms that is due to affinity.

Maxwell's law needs also modification when the gas is subject to external forces, such as gravity. We may here neglect this action, as it does not come into account in physical researches, but only in meteorological investigations; it is therefore sufficient to mention that, in addition to Maxwell himself, there are, among others, Boltzmann,2 Loschmidt,3 and Ferrini who have occupied themselves with this extension of Maxwell's law.

26. Fuller Explanation of Maxwell's Law

According to Maxwell's law the occurrence of the zero value for one of the three components of velocity is more frequent than that of any other given value. One might be inclined, therefore, to conclude that the most frequently occurring case would be that in which each component, and consequently the resultant velocity, is vanishingly small. This conclusion, however, must be false, as it can only extremely seldom happen that a molecule comes to rest in the midst of a swarm of molecules rushing rapidly about.

It is easy to explain the apparent contradiction between Maxwell's law and this undoubtedly true fact. When one of the components is zero there is no necessary reason for the other two to vanish, but they may have any possible value. In the case, therefore, which, according to Maxwell's theory, is the most probable, one of the three components may very well vanish without the resultant

B. A. Report, 1873, p. 29.

2 Wiener Sitzungsber. Ixxii. 2. Abth. 1875, p. 427.

3 lbid. lxxiii. 2. Abth. 1876, pp. 128, 366.

Rendiconti d. R. Istituto Lombardo [2] xviii. 1885, p. 319.

E

velocity of the molecule being zero. We shall rather obtain a value differing from zero in determining the mean of the absolute values of the molecular speed without reference to direction, in all those cases in which one of the three components is zero.

Maxwell's law must therefore be differently expressed than it is in § 24, when it has reference not to the components of the velocity, but to the actual speed itself. We can then explain the law as follows:

There is a most probable value of the speed which occurs more frequently than any other. Other values of the speed, whether greater or less, occur the oftener or are the more probable the nearer they approach to equality with the most probable. Infinitely great as well as infinitely small values of the speed have infinitely small probability. Molecules at rest, therefore, are infinitely seldom to be met with.

The connection between the two different forms of the law may be clearly illustrated by a comparison employed by Maxwell in a lecture.

If practised marksmen shoot at a target, the hits are most crowded together near the centre, and there are but few shots near the edge; the marks are approximately represented by the figure on page 51. In this case, too, the distribution of the hits follows the same law which Maxwell has found for the molecules of gas, viz. the law of errors; for a shot at the target is an attempt to hit the centre, just as a measurement is an attempt to hit the true value of the measured magnitude. Small deviations from the centre are therefore more probable than large ones in target practice also. The shots can deviate to right or left, above or below, and thus both horizontally and vertically. We have therefore in this case two components of deviation to distinguish, a horizontal and a vertical; and for each component the value zero is the most probable, since if we draw a series of parallel lines, say vertically, on the target, that one which passes through the centre passes also through more shotmarks than any other. But what is true for the components

On the Dynamical Evidence of the Molecular Constitution of Bodies,' J. Chem. Soc. xiii. 1875, p. 438 ; Nature, xi. p. 357; Scientific Papers, ii. p. 418.

cannot be immediately transferred to the resultant deviation from the centre. The points that are equally distant from the centre do not lie on a straight line, but on a circle described about the centre; and the circle which passes. through the most shot-marks is by no means one of the innermost circles of the target, for the inner rings are too small to contain many marks-in fact, the circle passing through the most shot-marks lies on a ring of medium size. The same relations hold also for the values of the speeds

FIG. 1

and of the components of velocity which occur the most frequently among the molecules of gas.

A more exact representation of the unequal probability of different values of the speed is given by the curve on page 52, the course of which shows us the law regulating the number of molecules which move with a given speed. In the figure the function which determines the frequency of a value is represented by a curve with the equation

This construction means that the magnitude of the ordinate y of the curve is equal to the probability of a speed whose

magnitude is equal to that of the abscissa x, and for the unit of speed that value is chosen which is the most probable.

This graphic representation of the law lets us easily see that the values of the speed that occur with any considerable frequency are only slightly different from that of greatest probability, whence we might conclude that the idea of all the molecules possessing equal speed is really approximately admissible. For a speed which is three or even only two

and a half times as great as the most probable speed has an almost vanishing probability, as a glance at the figure shows, so that no speeds can in fact occur which considerably surpass this value. And this is the case, too, with markedly smaller speeds.

Pirogoff has therefore gone so far as to assume that the values of the molecular speed which really occur lie between fixed limits, both the very large and the very small values of the speed being cut out by some equalising action such as a resistance or something of that kind. We may, in fact, as we easily see from the curve, determine such

1 Fortschr. d. Physik, 1886, 42. Jahrg. 2. Abth. p. 241.