438 ON boscovich's theory. 



through the six tetrahedral angles of the dodekahedrou arouad O, we 

 tind a homogeneous equilateral distribution of six-atom groups, in equi- 

 librium. 



Place now an atom at each point O- The equilibrium will be dis- 

 turbed in each case, but there will be equilibrium with a different 

 value of r (still between o and a). Thus we have nine-atom groups and 

 seven atom groups. 



Thus in all, we have found homogeneous distributions of six-atom, of 

 seven-atom, of eight-atom, and of nine-atom groups, each in equilibrium. 

 Without stopping to look for more complex groups, or for live-atom, or 

 four-atom groups, we find a homogeneous distribution of three-atom 

 groups in equilibrium by placing an atom at every point O, and at each 

 of the eight points A], As, Aj, Ac, A.j, A7, A4, A;(. Thus we see by ob- 

 serving that each of these eight A's is common to four tetrahedrons of 

 A's, and is in the center of a tetrahedron of O's; because it is a common 

 trihedral corner point of four contiguous dodekahedrons. 



Lastly, choosing A2, A3, A4, so that the angles A1OA2, A1OA3, A1OA4, 

 areeach obtuse,* we makeahomogeneousassemblageof two atom groups 

 in equilibrium by placing atoms at O, Aj, A2, A3, A4. There are four 

 obvious ways of seeing this as an assemblage of di-atomic groups, one 

 of which is as follows: Choose Aj and O as one pair. Through A2, A3, 

 A4 draw lines same-wards parallel to AiO, and each equal to AiO. 

 Their ends lie at the centers of neighboring dodekahedrons, which pair 

 with A2, A3, A4, respectively. 



For the Boscovich theory of the elasticity of solids, the consideration 

 of this homogeneous assemblage of double atoms is very important, 

 liemark that every O is at the center of an equilateral tetrahedron of 

 four A's J and every A is at the center 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 neigh- 

 bors; and (J and three of its twelve nearest O neighbors. 



[By aid of an illustrative model showing four of the one set of tetra- 

 hedrons with their corner atoms i^ainted blue, and one tetrahedron of 

 atoms in their centers, painted red, the mathematical theory which the 

 author had communicated to the Eoyal Society of Edinburgh, was illus- 

 trated to section A.] 



In this theory it is shown that in an elastic solid constituted by a 

 single homogeneous assemblage of Boscovich atoms, there are in gen- 

 eral two diftereut rigidities, «, %i, and one bulk-modulus. A-; between 

 which there is essentially the relation ok = 3u + 2»,, whatever be the 

 law of force. The law of force may be so adjusted as to make ni = n ; 

 and in this case we have 3/i- = on, which is Poisson's relation. But no 

 such relation is obligatory when the elastic solid consists of a homoge- 



* This also makes A2OA3, A2OA4, and A3OA4 eacb obtuse. Each of these six obtuse 

 angles is efpial to 180 — cos -'(i)- 



