ON BOSCOVICH'S THEORY.*

By Sir WILLIAM THOMPSON.

Without accepting Boscovich's fundamental doctrine that the ultimate atoms of matter are points endowed each with inertia and with mutual attractions or repulsions dependent on mutual distances, and that all the properties of matter are due to equilibrium of these forces, and to motions, or changes of motion produced by them when they are not balanced, we can learn something towards an understanding of the real molecular structure of matter, and of some of its thermo-dynamic properties, by consideration of the static and kinetic problems which it suggests. Hooke's exhibition of the forms of crystals by piles of globes, Naviers' and Poisson's theory of the elasticity of solids, Maxwell's and Clausins' work in the kinetic theory of gases, and Tait's more recent work on the same subject-all developments of Boscovich's theory pure and simple-amply justify this statement.

Boscovich made it an essential in his theory that at the smallest distances there is repulsion, and at greater distances attraction; ending with infinite repulsion at infinitely small distance, and with attraction according to Newtonian law for all distances for which this law has been proved. He suggested numerous transitions from attraction to repulsion, which he illustrated graphically by a curve-the celebrated Boscovich curve-t explain cohesion, mutual pressure between bodies in contact, chemical affinity, and all possible properties of matter-except heat, which he regarded as a sulphureous essence or virtue. It seems now wonderful that after so clearly stating his fundamental postulate which included inertia, he did not see inter-molecular motion as consequence of it, and so discover the kinetic theory of heat liquids, and gases; and that he only used his inertia of the or ass explain the known phenomena of the inertia of palpable masses, emblages of very large numbers of atoms.

a necessary

for solids atoms to

to A communication to Section A of the British Association A. S., at Newcastle, Septmber 13, 1889. (Report of the British Association, vol. LIX, pp. 494-196. Also, Naure, October 3, 1889, vol. XL, pp. 545-547.)

435

436

It is also wonderful how much towards explaining the crystallogra phy and elasticity of solids, and the thermo-elastic properties of solids, liquids, and gases, we find without assuming more than one transition from attraction to repulsion. Suppose for instance the mutual force between two atoms to be repulsive when the distance between them is< Z; zero when it is = Z; and attractive when it is > Z; and consider the equilibrium of groups of atoms under these conditions.

A group of two would be in equilibrium at distance Z, and only at this distance. This equilibrium is stable.

A group of three would be in stable equilibrium at the corners of an equilateral triangle, of sides Z; and only in this configuration. There is no other configuration of equilibrium except with the three in one line. There is one, and there may be more than one, configuration of unstable equilibrium, of the three atoms in one line.

The only configuration of stable equilibrium of four atoms is at the corners of an equilateral tetrahedron of edges Z. There is one, and there may be more than one configuration of unstable equilibrium of each of the following descriptions:

(1) Three atoms at the corners of an equilateral triangle, and one at its center.

(2) The four atoms at the corners of a square.

(3) The four atoms in one line.

There is no other configuration of equilibrium of four atoms, subject to the conditions stated above as to mutual force.

In the oral communication to Section A, important questions as to the equilibrium of groups of five, six, or greater finite numbers of atoms were suggested. They are considered in a communication by the author to the Royal Society of Edinburgh, of July 15, to be published in the Proceedings before the end of the year. The Boscovichian foun dation for the elasticity of solids with no inter-molecular vibrations was slightly sketched, in the communication to Section A, as follows:

Every infinite homogeneous assemblage Boscovich atoms is in equilibrium. assemblage, pro vided that extraneous forces be applied to all w

So therefore is every finite homoge muential dis

tance of the frontier, equal to the forces which a honi eneous continuation of the assemblage through influential distance bey the frontier would exert on them. The investigation of these extrains forces for any given homogeneous assemblage of single atoms, or roups of atoms, as explained below, constitutes the Boscovich equilibeory

of elastic solids.

To investigate the equilibrium of a homogeneous assemblage

***Homogeneous assemblage of points, or of groups of points, or of bodies, or of sys bodies,” is an expression which needs no definition, because it speaks for itsell ambiguously. The geometrical subject of homogeneous assemblages is treated w perfect simplicity and generality by Bravais, in the Journal de l'Ecole Polytechni

cahier xix, pp. 1-128. (Paris, 1850.)

or more atoms, imagine in a homogeneous assemblage of groups of i atoms, all the atoms except one held fixed. This one experiences zero resultant force from all the points corresponding to it in the whole assemblage, since it and they constitute a homogeneous assemblage of single points. Hence it experiences zero resultant force also from all the other i-1 assemblages of single points. This condition, fulfilled for each one of the atoms of the compound molecule, clearly suffices for the equilibrium of the assemblage, whether the constituent atoms of the compound molecule are similar or dissimilar.

When all the atoms are similar-that is to say, when the mutual force is the same for the same distance between every pair-it might be supposed that a homogeneous assemblage, to be in equilibrium, must be of single points; but this is not true, as we see synthetically, without reference to the question of stability, by the following examples, of homogeneous assemblages of symmetrical groups of points, with the condition of equilibrium for each when the mutual forces act.

Preliminary.-Consider an equilateral homogeneous assemblage of single points, O, O', etc. Bisect every line between nearest neighbors by a plane perpendicular to it. These planes divide space into rhombic dodekahedrons. Let A,OA5, A20A6, AОA, AOA, be the diagonals through the eight trihedral angles of the dodekahedron inclosing O, and let 2a be the length of each. Place atoms Q1, Q5, Q2, Q6, Q3, Q7, Q4. Q8, on these lines, at equal distances, r, from O; and do likewise for every other point, O', O", etc., of the infinite homogeneous assemblage. We thus have, around each point A, four atoms, Q. Q', Q", Q"", contributed by the four dodekahedrons of which trihedral angles are contiguous in A, and fill the space around A. The distance of each of these atoms from A is a - r.

Suppose, now, r to be very small. Mutual repulsions of the atoms of the groups of eight around the points O will preponderate. But suppose a-r to be very small: mutual repulsions of the atoms of the groups of four around the points A will preponderate. Hence for some value of r between O and a, there will be equilibrium. There may (according to the law of force) be more than one value of r between O and a giving equilibrium; but whatever be the law of force, there is one value of r giving stable equilibrium, supposing the atoms to be constrained to the lines OA, and the distances r to be constrainedly equal. ms, or is clear from the symmetries around O and around A, that neither of equilibise constraints is necessary for mere equilibrium; but without them

om

bey

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assemblage

equilibrium might be unstable. Thus we have found a homogenequilateral distribution of eight-atom groups, in equilibrium. Simi

bodies, or of syy placing atoms on the three diagonals, B1OB4, B2OB5, BзOB

speaks for itself ages is treated wi

"Ecole Polytechniqu

cans such an assemblage as that of the centers of equal globes piled homoas in the ordinary triangular-based, or square-based, or oblong-rectanglemids of round shot or of billiard balls.

through the six tetrahedral angles of the dodekahedron around O, we find a liomogeneous equilateral distribution of six-atom groups, in equilibrium.

Place now an atom at each point O. The equilibrium will be disturbed in each case, but there will be equilibrium with a different value of r (still between o and a). Thus we have nine-atom groups and

seven atom groups.

Thus in all, we have found homogeneous distributions of six-atom, of seven-atom, of eight-atom, and of nine-atom groups, each in equilibrium. Without stopping to look for more complex groups, or for five-atom, or four-atom groups, we find a homogeneous distribution of three-atom groups in equilibrium by placing an atom at every point O, and at each of the eight points A1, A5, A2, A6, A3, A7, A4, As. Thus we see by observing that each of these eight A's is common to four tetrahedrons of A's, and is in the center of a tetrahedron of O's; because it is a common trihedral corner point of four contiguous dodekahedrons.

Lastly, choosing A2, A3, A4, so that the angles A¡OA2, A¡OAз, A¡OA4, are each obtuse,* we make a homogeneous assemblage of two-atom groups in equilibrium by placing atoms at O, A1, A2, A3, A4. There are four obvious ways of seeing this as an assemblage of di-atomic groups, one of which is as follows: Choose A, and O as one pair. Through A2, A3, A, draw lines same-wards parallel to A,O, and each equal to A,O. Their ends lie at the centers of neighboring dodekahedrons, which pair with A2, A3, A4, respectively.

For the Boscovich theory of the elasticity of solids, the consideration of this homogeneous assemblage of double atoms is very important. Remark that every O is at the center of an equilateral tetrahedron of four A's; and every A is at the center of an equal and similar, and same ways oriented, tetrahedron of O's. The corners of each of these tetrahedrons are respectively A and three of its twelve nearest A neighbors; and O and three of its twelve nearest O neighbors.

[By aid of an illustrative model showing four of the one set of tetrahedrons with their corner atoms painted blue, and one tetrahedron of atoms in their centers, painted red, the mathematical theory which the author had communicated to the Royal Society of Edinburgh, was illus trated to section A.]

In this theory it is shown that in an elastic solid constituted by a single homogeneous assemblage of Boscovich atoms, there are in general two different rigidities, n, n, and one bulk-modulus, k; between which there is essentially the relation 3k3n+ 2n1, whatever be the law of force. The law of force may be so adjusted as to make n1 = n; and in this case we have 3k = 5n, which is Poisson's relation. But no such relation is obligatory when the elastic solid consists of a homoge

*This also makes A¿OAз, A ̧OA, and à ̧OA, each obtuse. Each of these six obtuse angles is equal to 180 cos (†).