546 



NATURE 



\Oct. 3, 1889 



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 fol- 

 lowing descriptions : — 



(i) Three atoms at the corners of an equilateral tri- 

 angle, and one at its centre. 



(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 verbal communication to Section A, inportant 

 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 foundation 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^ of Boscovich 

 atoms is in equilibrium. So, therefore, is every finite 

 homogeneous assemblage, provided that extraneous forces 

 be applied to all within influential distance of the frontier, 

 equal to the forces which a homogeneous continuation of 

 the assemblage through influential distance beyond the 

 frontier, would exert on them. The investigation of these 

 extraneous forces for any given homogeneous assemblage 

 of single atoms — or of groups of atoms as explained below 

 — constitutes the Boscovich equilibrium-theory of elastic 

 solids. 



To investigate the equilibrium of a homogeneous 

 assemblage of two or more atoms, imagine, in a homo- 

 geneous assemblage of groups of i atoms, all the atoms 

 except one held fixed. This one experiences zero result- 

 ant force from all the points corresponding to it in the 

 whole assemblage, since it and they constitute a homo- 

 geneous assemblage of single points. Hence it experi- 

 ences zero resultant force also from all the other i - \ 

 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 homo- 

 geneous 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 fol- 

 lowing examples, of homogeneous assemblages of sym- 

 metrical groups of points, with the condition of equilibrium 

 for each when the mutual forces act. 



Prelimi7iary. — Consider an equilateral ^ homogeneous 

 assemblage of single points, O, O', &c. Bisect every 

 line between nearest neighbours by a plane perpen- 

 dicular to it. These planes divide space into rhombic 

 dodekahedrons. Let A1OA5, AjOAg, AgOA,., A4OA8, 



' ''^Homogeneous assemblage of joints, or of groicps of foints, or of 

 lociies, or of systems of bodies," is an expression which needs no definition, 

 because it speaks for itself unambiguously. The geometrical subject of 

 hoiTiogeneous assemblages is treated, with perfect simplicity and generality 

 ly Bravais, in x^ci^ Journal de r Ecole Polytechnigue, caSier ja-x.. x>V- 1-128 

 (faris, 1850). 



_"' 'J his means such an assemblage as that of the centres of equal globes 

 piled homogeneously, as in the ordinary trian?ular-based, or square-based, 

 or cbljng-rectangle-based, pyramids of round shot or of billiard balls. 



be the diagonals through the eight trihedral angles 

 of the dodekahedron inclosing O, and let 2a be the 

 length of each. Place atoms Qi, Qg, Q2, Qb, Q3, 

 Q?) Q4) Qs) on these lines, at equal distances, r, 

 from O ; and do likewise for every other point, O', O", 

 &c., of the infinite homogeneous assemblage. We thus, 

 have, around each point A, four atoms, O, O', O", Q"V 

 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. It is clear from the symmetries around O and 

 around A, that neither of these constraints is" necessary 

 for mere equilibrium ; but without them the equilibrium 

 might be unstable. Thus we have found a homogeneous 

 equilateral distribution of 8-atom groups, in equilibrium. 

 Similarly, by placing atoms on the three diagonals, 

 B1OB4, BoOB-. BgOBe, through the six tetrahedral angles 

 of the dodekahedron around O, we find a homogeneous 

 equilateral distribution of 6-atom groups, in equilibrium. 



Place, now, an atom at each point O. The equilibrium 

 will be disturbed in each case, but there will be equi- 

 librium with a different value of r (still between o and a). 

 Thus we have 9-atom groups and 7-atom groups. 



Thus, in all, we have found homogeneous distributions 

 of 6-atom, of 7-atom, of 8-atom, and of 9-atom groups, 

 each in equilibrium. Without stopping to look for more 

 complex groups, or for 5-atom or 4-atom groups, we find 

 a homogeneous distribution of 3 atom groups in equi- 

 librium by placing an atom at every point O, and at each 

 of the eight points Ai, Ag, Ao, Ag, A3, Ay, A4, As. This 

 we see by observing that each of these eight A's is com- 

 mon to four tetrahedrons of A's, and is in the centre of a 

 tetrahedron of O's ; because it is a common trihedral 

 corner point of four contiguous dodekahedrons. 



Lastly, choosing Ag, A3, A4, so that the angles AjOAj, 

 A1OA3, A^OAj are each oljtuse,^ we make a homo- 

 geneous assemblage of 2-atom groups in equilibrium by 

 placing atoms at O, Aj, Ag, A3, A4. There are four ob- 

 vious ways of seeing this as an assemblage of di-atomic 

 groups, one of which is as follows : — Choose Ai and O as 

 one pair. Through Ag, A3, A4 draw lines same-wards 

 parallel to AjO, and each equal to AjO. Their ends lie 

 at the centres of neighbouring dodekahedrons, which pair 

 with Ao, 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 

 centre of an equilateral tetrahedron of four A's ; and 

 every A is at the centre 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 neighbours ; and O and three of 

 its twelve nearest O neighbours. 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 centres, painted red, the mathematical 

 theory which the author had communicated to the Royal 

 Society of Edinburgh, was illustrated to Section A. 



In this theory it is shown that in an elastic solid con- 

 stituted by a sinjle homogeneous assemblage of Boscovich 

 atoms, there are in general two different rigidities, «, n^, and 



' This also makes A2OA3, A2OA4, and A3OA4 each obtuse. Each of 

 these six obtuse angles is tqual to 180 -cos "iCi's)- 



