ON boscovich's theory. 437 



or more atoms, imagine in a homogeneous assemblage of gronps of ?* 

 atoms, all the atoms excei)t one held fixed. This one experiences zero 

 resnltant force from all the points corresponding to it in the whole as- 

 semblage, since it and they constitnte a homogeneous assemblage of 

 single points. Hence it experiences zero resultant force also from all 

 the other i—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 bet^Yeen every pair — it might be sup- 

 posed that a homogeneous assemblage, to be in equilibrium, must be of 

 single points; but this is not true, as we see synthetically, without ref- 

 erence to the question of stability, by the following examples, of homo- 

 geneous assemblages of symmetrical groups of points, with the condition 

 of equilibrium for each wlien the mutual forces act. 



Preliminary. — Consider an equilateral* homogeueons 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 AiOA.s, A2OA,,, A:jOA7, A40Aii, be the diagonals 

 through the eight trihedral angles of the dodekahedron inclosing O, and 

 let 2a be the length of each. Place atoms Q,, Q5, Q^, Qe, Qs, Q75 Q^* Qs, 

 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, arouud 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 « — 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 sup- 

 pose 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 «, there will be equilibrium. There may (ac- 

 cording 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 con- 

 strained to the lines OA, and the distances r to be constrainedly equal. 

 It is clear fronj 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 homogene- 

 ous equilateral distribution of eight-atom groups, in equilibrium. Simi- 

 larly, by placing atoms on the three diagonals, B|OB.„ BaOB:,, B3OB6 



*Thi8 mean.s such an assemblage aa that of the centers of equal globes pihid homo- 

 geneously, as in the ordinary triangular-ba.seil, OT square-based, or oblong-rectangie- 

 based pyramids of round shot or of billiard balls. 



