clearly from a memoir by Gustav Lübeck,' who investigates the laws of the collision of massive particles without introducing any further suppositions than the above-named theorems. It follows that these assumptions are sufficient to determine the velocities with which the colliding particles separate from each other after an encounter when the velocities which the particles possessed before the encounter are known. We easily see, for instance, without calculation for direct impact, that from the two propositions of the conservation of kinetic energy and of the conservation of the motion of the centre of gravity we obtain two equations, which must be sufficient to fully determine the two unknown values of the velocities after the impact. The same thing holds in the more general case of oblique impact, in which only the number of unknown magnitudes is greater since several components come into

account.

We may from this conclude that the validity of the laws which determine the change of the motions on the occurrence of an elastic impact is not confined to the case of elastic bodies, but presupposes only that the above-named theorems of mechanics hold good. We may therefore assume that the laws of elastic impact are valid for the encounters of molecules also without thereby ascribing elasticity to the molecules themselves.

If according to this the answer to the question, what forces are developed at an encounter of two gaseous molecules, does not touch the foundation of the kinetic theory, that question still remains so important and interesting that we cannot pass it over in silence.

In memoirs upon this theory, and especially in the older ones, we often meet with the assumption that two molecules behave during an encounter like two elastic bodies, or even like two elastic spheres. This hypothesis has much that is tempting about it from the ease with which it can be handled; but the difficulty would only be transferred, and not overcome, if we proposed to explain the elasticity of gases by the elasticity of the molecules.

Schlömilch's Zeitschr. f. Math. u. Physik, 1877, 22. Jahrg. p. 126.

In order, therefore, to explain the behaviour of the gaseous particles, we must have recourse to the assumption of forces which the particles exert on each other, at least when near enough; and there arises only the question whether these forces are attractive or repulsive; for the latter may also come into play when the distance becomes small enough, though the gaseous molecules certainly attract each other at their mean distances of separation.

On this account Maxwell, too, for a long time defended the assumption of repulsive forces between gaseous particles. These repulsive forces were supposed to decrease more rapidly than the attractive forces as the distance increased, and, indeed, inversely in proportion to the fifth power of the distance, while the attractive forces were to be taken as following Newton's law of being inversely propɔitional to the square of the distance. Maxwell' arrived at this hypothesis, which allows of a very easy and elegant treatment, because the law of dependence of the viscosity of a gas on the temperature, which is deduced from this hypothesis, is that which he obtained by experiment. But since this law, viz. that the coefficient of viscosity of a gas is proportional to the absolute temperature, is contradicted by later observations, the hypothesis must be given up.

Other physicists have professed the view that attractive forces are to be assumed as acting between gaseous molecules. That such forces must be assumed unconditionally has been proved by Clausius by means of his proposition of the Virial, from which it follows that a stationary state of motion can be permanently maintained only if forces maintain the equilibrium dynamically, and that therefore a stable state is quite impossible without attractive forces. Further, Boltzmann3 has shown that it is also sufficient to assume only attractive forces, and no repulsive forces, between gaseous molecules. By assuming that strongly

1 Phil. Trans. 1867, clvii. p. 49; Phil. Mag. 1868 [4] xxxv. p. 129; Scient. Papers, ii. p. 26.

2 Bull. Ac. Belg. 1886 [3] xi. p. 193; Kinetische Theor. d. Gase, 1889-91, p. 264.

p. 37.

Wien. Sitzungsber. 1884, lxxxix. Abth. 2, p. 714; Wied. Ann. 1885, xxiv.

2

attractive forces act only at quite small distances, he obtains the very same laws for an encounter which Maxwell obtained with his repulsive forces. This result is in excellent agreement with Lübeck's proposition, of which we have made mention. We may therefore consider all the forces as attractive which gaseous molecules exert upon each other, whether at large or small distances apart. This corresponds to the experimental results of Joule and Lord Kelvin as to the heat-effects of flowing gases, and also no less to the ideas employed by Sutherland to explain the large variation of the coefficient of viscosity with the temperature.

The law of variation of the strength of these attractive forces with the distance between the particles cannot yet be decided from our present knowledge. It is, however, quite possible to calculate, or at least to estimate, the amount of the energy that comes into play during the action of these unknown forces. This is just as possible as it was possible to estimate the energy of the heat contained in a body without its being necessary to know the law of the molecular movement in which heat consists.

Boltzmann' has made a first essay in this direction, and has thereby proved that we have to do with forces of extraordinarily great intensity. In this calculation Boltzmann introduces manifold hypothetical suppositions which are approximately admissible for every case; we may especially mention that, just as we did in former paragraphs, he looks upon the molecules themselves as the same in the liquid and gaseous states, and upon their motions only as different.

With Boltzmann we will first of all calculate the amount of energy needed for two molecules of water, whose mean distance apart is p, to approach each other by the length xp, so that their distance apart is reduced to (1-x)p. For this we make use of a result of Grassi's experiments on the compressibility of liquids, viz. that water is compressed by 0.000048 of its volume by the pressure of one atmosphere, so that by the addition of an atmosphere to Wien. Sitzungsber. 1872, lxvi. Abth. 2, p. 213.

the pressure the distance apart p of two molecules is diminished by 0·000016p. An atmosphere is the pressure of 10334 kilograms per square metre, and it therefore exerts on the area p2 corresponding to a single molecule the force 10334 gp2 absolute units, wherein the kilogram is taken as unit of mass and the metre as unit of length, g being the acceleration of gravity. Since this force diminishes p by 0-000016p, the force required to diminish p by ap is

10334gp2x/0-000016,

if we may make the assumption (which is doubtless approximately correct) that the force is directly proportional to the amount of approach. The work done by this force while the molecules are approaching each other is found by multiplying the force by half the diminution of distance, and it is therefore

10334gp3x2/0-000032.

Boltzmann compares this expenditure of work with the energy of the molecular motion in water-vapour, in order to determine the shortening of distance, measured by x, which occurs at a collision of two molecules. Let m be the mass of a molecule, and therefore, with a kilogram and a metre as units, m = 1000p3, since 1 cubic metre of water weighs 1000 kilograms; further, with our former notation, wherein G represents the mean value of the molecular speed as calculated from the energy, the sum of the kinetic energies of the two colliding molecules is

where by § 28 the value of G for water-vapour at the temperature 0° is to be taken as 614 metres per second.

If we equate the value of the energy so determined to the above value found for the expended work, and also put g = 9.81, we have

,0-000032 / ?9.81px × 10334 = 3م6142 × 1000

According to this calculation the centres of two molecules of water-vapour at 0°, if they undergo direct collision when moving with their mean kinetic energy, approach to a distance of p, which is only two-thirds of their separation when they are in the liquid state at maximum density at 4°. In order to compress water so strongly, a pressure of 20,000 atmospheres must be employed.

Though several points in this calculation may have only a very doubtful justification, yet from the result we learn that the molecular forces which are developed on the encounter of two molecules possess a very considerable intensity.

From the value of the molecular energy we can at once form an estimate of the energy of the motion of the atoms within the molecule; for the ratio in which these two energies stand to each other is (§ 53) determined by the ratio of the two specific heats. The forces, therefore, to which the atoms are subject must also be in general very great.

The only exception is that of the case when the molecule is monatomic. In this case we might assume that there is not interior motion at all in the molecule, since it is simple and consists of but one atom. But this assumption cannot be right, for monatomic gases and vapoursmercury vapour, for instance-can radiate light the spectrum of which consists of a series of bright lines, and therefore internal motions must be present in a gaseous molecule that contains but one atom just as in a polyatomic molecule.

Eilhard Wiedemann' has cleared up this apparent contradiction by measuring the energy necessary to make a vapour luminous. He compared the light radiated by sodium vapour with that coming from a platinum wire made to glow by the passage of an electric current; from the resistance of the wire and the strength of the current he could determine the luminous energy in heat units, and

1 Wied. Ann. 1889, xxxvii. pp. 241, 248.