43 G ON BOSCOVICHS TUE014Y. 



It is also wonderful bow much towards explaiuiug 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* of Boscovich atoms is in 

 equilibrium. So therefore is every finite homogeneous assemblage, pro- 

 vided that extraneous forces be applied to all within influential dis- 

 tance of the frontier, equal to the forces Avhich a homogeneous continu- 

 ation 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 



'"■•Romof/eneous assemblage ofjmnts, or of groups of points, or of bodies, or of systems of 

 bodies,'' is an expression which needs no definition, because it speaks for itself un- 

 anibiguonsly. The geometrical subject of homogeneous assemblages is treated with 

 ])erfect simplicity and generality by Bravais, in the Journal de VEcole Folytechuique, 

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



