TRANSACTIONS OE" SECTION A. 495 



of equilibrium except with the three in one line. There is one, and there may be 

 more than one, eonlifruration of unstable equilibrium, of the three atoms in one line. 

 The only contiguration 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, contiguration of unstable equilibrium of each of the following descriptions : — 



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



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



(3) The four F.toms in one line. 



There is no otlier configuration of equiUbrium of four atoms, subject to the con- 

 ditions stated above as to mutual force. 



In the oral communication to Section A, important questions as to the equili- 

 brium 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 tinite homogeneous assemblage, provided thr.t 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 dis- 

 tance 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 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 homo- 

 geneous assemblage of single points. Hence it experiences zero residtant force also 

 from all tlie other t - 1 assemblages of single points. This condition, fulfilled for 

 each one of the atoms of the compound molecule, clearly sufhces for the equi- 

 librium of tiie assemblage, whether the constituent atoms of the compound molecule 

 are similar or dissimilar. 



When all the atom.s are similar— that is to say, when the mutual force is the 

 same for the same distance between every pair — it might be supposed tliat 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 

 following examples, of homogeneous assemblages of symmetrical groups of points, 

 with the condition of equilibrium for each when the mutual forces act. 



rreliminan/. — Consider an equilateral ^ homogeneous assemblage of single 

 points, O, O', Sec. Bisect every line between nearest neighbours by a plane perpen- 

 dicular to it. These planes divide space into rhombic dodekahedrons. Let 

 A,OAj, AoOA^, A3OA,, A,OA^ be the diagonals through the eight trihedral angles 

 of the dodekahedron enclosing 0, and let 2a be the length of each. Place atoms 

 Qi> Q.-,) Qj. Q,;( Q31 Qt* Qi» Qg o" these lines, at equal distances, /•, from O ; 

 and do like\vise for every other point, 0', 0", &c., of the infinite homogeneous 

 assemblage. We thus have, around each point A, four atoms, Q, Q', Q", (J'", con- 

 tributed 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 .\ is « - r. 



Suppose, now, r to be very small. Mutual repulsions of the atoms of the 

 proups of eight around the points will preponderate. But suppose « - ;• to be 



> ' IIomogiTicovs assemhlarfc of paintx, or of grovps of points, or of bodies, or of 

 systems of hodiex,' is an expression which needs no definition, because it speaks for 

 itself unambiguously. The geometrical subject of homogeneous assemblages is 

 treated with perfect simplicity and generality by Bravais, in the Journal de I'Ecole 

 Polytechniqnc, cahier six., pp. 1-128 (Paris, 1850). 



* This means such an assemblage as that of the centres of equal globes piled 

 homogeneously, as in the ordinary triangular-based, or square-based, or oblong- 

 rectangle-based pyramids of round shot or of bilhard-balls. 



