698 Proceedings of Royal Society of Edinburgh. [sess. 
(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, sub- 
ject to the conditions stated above as to mutual force. 
Important questions as to the equilibrium of groups of five, six, 
or greater finite numbers, of atoms occur, but must be deferred. 
The Boscovichian foundation for the elasticity of solids with no 
inter-molecular vibrations is the subject of §§ 62-71 below. A few 
preliminary remarks here may be useful. 
§ 18. Every infinite homogeneous assemblage * of Boscovich atoms 
is in equilibrium. So, therefore, is every finite homogeneous assem- 
blage, provided that extraneous forces be applied to all within in- 
fluential 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 — con- 
stitutes the Boscovich equilibrium-theory of elastic solids. 
§ 19. 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 
itself in the whole assemblage, since it and they constitute a homo- 
geneous assemblage of single points. Hence it must experience 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. 
§ 20. 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 syntheti- 
cally, without reference to the question of stability, by the following 
* “ Homogeneous assemblage of points, 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 unambiguously. The geometrical subject of homogeneous assem- 
blages is treated with perfect simplicity and generality by Bravais, in the 
Journal de Vficole Poly technique, cahier xxxiii. pp. 1-128 (Paris, 1850). 
