700 Proceedings of Royal Society of Edinburgh. [sess. 
different value of r (still between zero and a). Thus we have 9-atom 
groups and 7-atom groups. 
§ 24. Thus, in all, we have found homogeneous distributions of 
6-atom, of 7-atom, of 8-atom, and of 9-atom groups, each in equilibrium. 
Without stopping to look for more complex groups, or for 5-atom 
or 4-atom groups, we find a homogeneous distribution of 3-atom* 
groups in equilibrium by placing an atom at every point 0, and at 
each of the eight points A 1? A 5 , A 2 , A 6 , A 3 , A 7 , A 4 , A 8 . There are 
four obvious ways of seeing this, found by choosing one or other 
of the four diagonals through trihedral angles referred to in § 21. 
Take, for example, A 1 OA 5 , and its congeners for all the dodeka- 
hedrons. These triplets include all the A’s. (Compare § 25 below.) 
§ 25. Lastly, choosing A 2 , A 3 , A 4 , so that the angles AjOA^ 
AjOAg, A 1 OA 4 , are each obtuse, f we make a homogeneous assem- 
blage of 2-atom | groups in equilibrium by placing atoms at O, A 1} 
A 2 , A 3 , A 4 . There are four obvious ways (compare § 24 above) of 
seeing this as an assemblage of di-atomic groups, one of which is as 
follows : — Choose A 4 and O as one pair. Through A 2 , A 3 , A 4 , draw 
lines same-wards parallel to A 4 0, and each equal to A 4 0. Their 
ends lie at the centres of neighbouring dodekahedrons, which pair 
with A 2 , A 3 , A 4 respectively. 
§ 26. For the Boscovich theory of the elasticity of solids, the 
consideration of this homogeneous assemblage of double atoms is 
very important. Remark that every 0 is at the centre of an equi- 
lateral tetrahedron of four A’s ; and every A is at the centre of an 
equal and similar, but contrary-ways oriented, tetrahedron of O’s. 
The corners of each of these tetrahedrons are respectively A, and 
three of its twelve nearest A-neighbours ; and 0 and three of its 
twelve nearest O-neighbours. By aid of an illustrative model 
showing four of the one set of tetrahedrons with their corner atoms 
painted blue, and one tetrahedron of atoms in their centres painted 
red, the mathematical theory which had been communicated to the 
Royal Society of Edinburgh, was illustrated to Section A of the 
British Association at its recent meeting in Newcastle. 
* This is the assemblage described in the footnote on § 71 below. 
t This also makes A 2 OA 3 , A 2 OA 4 , and A 3 OA 4 each obtuse. Each of these six 
obtuse angles is equal to 180° - cos -1 (l/3). 
+ This is the assemblage described in § 69 below, and used in §§ 67, 68, 70. j 
