Boscovich, indeed, goes so far as to regard the atom as a mere centre of force, the result of whose existence is that no two atoms or centres can approach each other within a certain distance, while other physicists regard the atomic volume as a distinct portion of space occupied by that atom to the exclusion of every other, and comprising within it matter ideally infinitely divisible, but the parts of which in fact never have been, and never can be, separated from each other. In this latter mode of viewing the subject, all the conclusions of mechanics which are based on the conception of the continuity and infinite divisibility of matter may be applied to the equilibrium or motion of each individual atom, the atomic theory merely introducing the additional hypothesis that, in fact, these persistent entities called atoms do exist, and that out of them all substances which affect our senses are constructed. The theory of universal gravitation requires us to believe in the existence of forces or actions between every portion of matter and every other portion, determinate in magnitude and direction, and such that, when on the infinitely divisible hypothesis the volumes of these portions are indefinitely diminished, these mutual forces are inversely proportional to the square of the distance between the portions (the distance between any two points, one in the volume of each portion, being in this case taken as the distance between the portions), and directly proportional to the products of the masses, or quantities of the two portions of matter,—such forces being regarded provisionally as ultimate facts, while inviting further analysis and explanation. Chemical and chemico-physical investigations indicate the existence of other actions between portions of matter, following other and for the most part unknown laws, and rapidly becoming inappreciable as the distance between the reacting portions is increased. All these hypotheses are to be retained on the hypothesis of discrete atoms as above enunciated, the mutual actions between atoms being the resultant of the actions between the various portions of their constituent matter. The volumes of the atoms are so small that, for any sensible distances apart, the line of the resultant mutual action between them may be taken as coincident with the line joining any point in the volume of one to any point in the volume of the other, but, for distances or parts comparable with the linear dimensions of the atoms, the size and shape of their bounding surfaces must be taken into consideration, and perhaps also the law of distribution of their constituent matter within that surface. In all respects, unless we accept the Boscovichian hypothesis, we simply regard the atom as made up, so to speak, of infinitely divisible matter, while substances, as we know them, are built up of indestructible and unchangeable atoms.

With this conception of an atom, as thus explained, we might be content to rest, confessing our total ignorance of the mode in which such atoms are built up into actual substances, being satisfied to regard such substances as composed of these distinct portions of matter separated, or capable of being separated, by empty space from other portions. But the molecular hypothesis of the constitution of different kinds of substances aims at analysing this process by which such substances are built up out of their constituent atoms. The molecule of any substance is, by some chemists, defined as being the smallest portion of that substance to which can be attributed all the chemical properties of the substance; by others, as the smallest portion which, so long as the substance is chemically unchanged, keeps together without complete separation of its parts. In the language of Clausius’s theorem, if the parts of the molecule have internal motion, the kinetic energy of such internal motion is equal to the virial of the mutual attractive forces of the parts. Thus the formation of the molecule of each particular substance is viewed as an essential step in the process of building up that substance out of its constituent atoms. The molecule is first built up out of atoms arranged in its formation according to a definite type, and then the substance itself is constituted of these molecules. Of course molecules may be, and in fact in many particular substances are, supposed to be monatomic; that is to say, the intermediate step of building up the molecule out of the atoms has, in these particular substances, been omitted, the atoms and molecules becoming then identical. The particular arrangement of the formed molecules in the building up of the substance determines the physical state of that substance,—that is, its fluid, solid, gaseous, crystalline, or amorphous state; but the chemical properties of the substance depend upon the constitution of the molecule. As the investigations and theories of chemistry appear to indicate irresistibly the existence of permanent atoms, so do they also lead almost as necessarily to the conception of the molecule as an entity which bears the same relation to special substances that the atoms bear to matter generally. So long as the molecule endures, the substance of which it is the molecule retains its chemical properties; with the dissolution of the molecule, the substance, as that special substance, perishes; the atoms alone continue, and are free to enter into other combinations. The permanence of the molecule is relative, that of the atom absolute. This conception of the molecular constitution of substances suggests physical questions of great interest, such as the shape, volume, and mass of the constituent molecules, and the relative motions of which their parts are susceptible; and the answers to these questions cannot fail to be of great value in chemical and chemico-physical investigations, as well as in the theories of light and electricity.

Now, whatever differences may exist between the properties of different substances in the solid and liquid states, there are certain properties which, in the gaseous state, manifest themselves with no variation whatever in all substances alike. Hence the explanation of these common properties—or gaseous laws, as they are called—has long possessed a peculiar fascination for physicists. The tendency to expand or fill all accessible space, manifested by all gases, proves that on the molecular hypothesis their compound atoms or molecules must be continually tending to fly apart. We must conceive gases as constituted of molecules, not only separable but actually separated by space void of the matter of which these gases consist; and it may be most reasonably expected, therefore, that any general laws to which substances in this state conform may afford us a valuable insight into the constitution of these separate molecules.

Now the general laws to which all gases conform are: (1) Boyle’s law— that, in a given mass of any gas kept at constant temperature, the pressure per unit of area upon the containing surface increases in the same proportion as the volume occupied by the gas is diminished, or at least with very slight deviation from exact proportionality · (2) Charles’s law — that, if the temperature be varied while the pressure upon the gas remains the same, the gas increases by τrsd of its volume at zero centigrade for every degree of centigrade added to the temperature, or, which in combination with Boyle’s law is the same thing, that if the density be constant, the pressure is directly proportional to the temperature measured from the point - 273° centigrade, this point being called the zero of absolute temperature; (3) Avogadro’s law — which asserts that all gases at the same temperature and pressure contain the same number of molecules in the same volume; and (4) Dalton’s law— that in a mixture of different gases, when there is equilibrium, each gas behaves as a vacuum to all the rest.

It was at one time considered that these phenomena might be explained on the hypothesis of mutual repulsive forces between the parts of which the gas is composed, whether they were regarded as constituted of molecules or of infinitely divisible continuous matter, but it has been shown in the article Atom (vol. iii. p. 39 sq.) that there are at least two absolutely conclusive reasons why this explanation cannot be accepted. These objections, together with the experimental fact proved by Joule that gases, or at any rate atmospheric air, expand into vacuum with scarcely any appreciable change of temperature, must be considered fatal to any mutual-force theory of gaseous action, and, accordingly, physicists have been driven to seek for other methods of explaining these laws. The explanation which has been more developed than any other is that known as the kinetic theory of gases, which regards the intrinsic energy of a gaseous mass as residing, not in the potential energy of intermolecular forces, but mainly in the kinetic energy of the molecules themselves, which are assumed to be in a state of continual relative velocity, admitting at the same time a possible small intermolecular potential energy, and it may be also an interatomic energy, between the atoms of the individual molecules. That some such persistent relative motion does exist in every gaseous mass is evident from the rapidity with which odours penetrate the stillest air where no breath of wind—that is, of absolute motion of translation of the mass as a whole or any portion of finite size—is perceptible. It becomes an interesting question whether the laws of mechanics admit of a mass thus constituted ever arriving at a state of permanence; that is to say, whether, consistently with the hypothesis of infinite irregularities in the directions and magnitudes of velocities of individual molecules, there may be found any properties of the mass in the aggregate which remain constant, and in agreement with the accepted laws common to all gases. Now the physical theory of heat compels us to regard the intrinsic energy of any gaseous mass as dependent entirely or almost entirely upon the temperature. If, therefore, this intrinsic energy is to be sought for in the kinetic energy of the moving molecules, it follows that the average value of the kinetic energy of the molecules taken throughout the mass must be also a function of the temperature.

We will proceed to investigate the condition of permanence of a number of molecules moving about irregularly in any bounded space; and, for simplicity’s sake, we shall first of all restrict ourselves to the case of monatomic molecules.

We know nothing of the size or shape of these atoms, except that the volume of each one must be incomparably smaller than that of the containing region. In shape we shall, as the simplest hypothesis, regard them as spherical. We shall suppose that there are no intermolecular forces between any two such atoms, except of such a nature as to be practically insensible when the atoms are not geometrically in contact, and similarly as regards the forces between the atoms and the material bounding surface, such forces being of the nature called “conservative.” So that in point of fact we are investigating the mechanical properties of an infinitely large number of infinitely small and perfectly elastic spheres moving about in a given region and subject to frequent collisions.

Problem. — A very large number of smooth elastic spheres, equal in every respect, are in motion within a region of space of given volume, and therefore occasionally impinge upon each other with various degrees of relative velocity and in various relative directions; required to find the law of distribution of velocities in order that such distribution may be permanent.

Let Λ r be the total number of spheres, and let χ (w, V, w) du dv dw be the number of spheres whose component velocities, parallel to the axes, are intermediate between u and u + du, v and v + dv, w and w + dw respectively.

If c be the resultant velocity of any of these last-mentioned spheres, and if θ be the inclination of c fo the axis of z, and φ that of the plane cz to the plane xz, the last-mentioned expression will become, by changes of the independent variables from x, y, z to θ, φ, and c, χ (u, v, w) c? sin θ dθ dφ de.

Let a spherical surface of radius unity be described about any origin as centre, and let dσ be written for the element sin θ dθ dφ on this surface, then the last-written expression becomes

χ (it, v, w) c 2 de dσ.

Since for the same magnitude of the resultant velocities all directions of motion must be equally probable, it follows that the coefficient of de dσ in the last-written expression must be a function of c only, and therefore the number of spheres having component velocities between u and u + du, v and v + dv, w and w + dw, must be φ (c) du dv dw.

It is required to find the form of ψ in order that the value of this expression may be unaffected by collisions. The solution is, that the number of spheres with component velocities between the limits w and u + du, v and v + dv, w and w ÷ dw must be

Je^ ftc2 du dv dw; or Ae~ hc2 c 2 de dσ, employing the notation already used.

Integrating with respect to dσ from 0 to 4ττ, we find for the number of spheres with velocities between c and c + dc the expression

4τr Ae~^ ιc2 de.

Again, since the number with component velocities between u and u + du, v and v + dv, w and w + dw is Ac - h < u2 + v * + w2 > du dv dw, or (JJA~e- hu2 du} (l∕Ae~ hυ2 dυ) (ξ∕Λe~ hv,i dw), it follows that the number of spheres having velocities intermediate between u and u + du parallel to the x axis is

∕∙OO fCO

Ae~ fιu2 du i e ^ hυ2 dv ι e~ iu ° 2 dw; J oo √ oo that is, ^-Ae ~ hu2 du, h where A is to be determined by the equations -^A I e~ fιu2 du—N, J 00

. τri vr or A -= r = N; hi j Nhi therefore A = —v-; π i that is to say, the number of spheres having velocities between c and c + de is

4λi -7⅛c a a -=- Ne de.

√τr

Multiplying this expression by c, and integrating the product with regard to c from 0 to ∞, and dividing by N, the mean velocity for all the spheres becomes

2

V πh ’ and multiplying by c 2 instead of by c, we find the mean square of all the velocities to be

3_

2Λ'

In the preceding investigation no account has been taken of collisions between the spheres and the enclosing boundary of the region in which they are contained, because in every such collision the magnitude of the velocity of each sphere is unaltered and its direction is changed according to the ordinary law of reflexion, whence it is evident that the distribution is unaffected by such collisions. Also, the investigation has been confined to the cases of spheres colliding in pairs, but since there need be no limit to the smallness of the interval between any pair of collisions the result really embraces the cases of simultaneous collisions between three or more spheres; for if a sphere A collides with another B, and immediately afterwards with a third C, the resultant velocity of A after this second collision must be the same as if it had collided with B and C simultaneously.

The foregoing investigation has been given in some detail because the principles upon which it proceeds are essentially the same as those by which all questions of the distribution of energy among a great number of moving bodies are determined, although it may be found, as well as the detailed investigations of the results immediately following, in published memoirs and systematic treatises on the kinetic theory of gases.

If the spheres be not all of equal mass, but if there be within the region N spheres of mass m, N , of mass m', and so on, then it may be proved, by reasoning exactly similar to the foregoing, that when the permanent or stable state of motion has been attained the number of spheres of the N set with component velocities between u and u + du, v and v + dv, w and w + dw is

λmc 2 Ae 2 du dv dw, and the number of the N' set having component velocities between u' and u' + du', d and √ + dd, w and w + du∕, is

⅛m'c' 2 A'e 2 du dv' dw', where c 2 =m 2 + v 2 + w 2 , c' a =u' 2 + t∕ 2 + w 8 , h is a constant the same for both sets, and . N ∕ mh∖i N' ∕ ’ m'h∖i and so on if there be any other sets.

The mean velocity and mean square velocity of each sphere of the N set are 2 ∕^ 2 ~ j 3 .. 1

—7=. ∕ — τ and — r respectively, a v 7Γ z ∖∕ 77l∕i' ’ )TUl> and the mean kinetic energy of each of such spheres is 2h ’ the last result being common to all the sets.

If the spheres in the given region be acted on by any given forces tending to fixed centres, and functions of the distances of the centres of the spheres from the centres of force, we may not in such case assume, a priori, that the chances of velocities in all directions are the same; but we may assume that the number of spheres of any set (Λ 7 ) with coordinates of their centres intermediate between X and x + dx, y and y + dy, z and z + dz, and component velocities intermediate between u and u + du, v and υ + dv, w and w + dw, is ψ (x, y, z, u, v, w) dx dy dz du dv dw.

In the state of permanence the form of ψ must be independent of the time (t), so long as the sphere is moving free from collisions with any other.

From the last-mentioned condition it must follow that, if ≠ 1 =α 1 , ≠ 2 = α 2 , &c., be any equations among the variables determining the position and motion of any sphere obtained by the elimination of t from the equations of motion of that sphere, then ψ must be of the form ψ (≠j, φ. i , &c. ). With the assumption, then, that the number of spheres of the given set with variables between the above-mentioned limits is

≠ (≠ι, ≠ 2 ∙∙∙) dx...dw, we find for the form of φ, by reasoning like the foregoing, ⅛ ( I m ° 2 ∖

Ae × 2 ', where χ is the potential energy of the sphere in the position x, y, z, and cP=u 2 + v 2 + w 2 , and h is a constant, the same for all the sets.

j i ∕ mc 2 ∖

If we integrate the expression Ae × 2 'dx dy dz du dv dw for all values of x, y, z within the given region, we find for the number of spheres of any set with component velocities between u and u + du, v and v + dv, w and w + dw, hmc 2 Be 2 du dv dw, whence we easily see that the chances of velocities in all directions are the same, and that the mean velocity and mean square velocity of any sphere of this set are and respectively, and the mean

3

kinetic energy of any such sphere is and therefore the same for all the sets.

Furthermore, if we integrate the expression

-A (χ+^)

Ae ' 2 'dx dy dz du dv dw for all values of u, v, and to from — co to + ∞ respectively, we obtain...