496 REPORT— 1889. 



very small: mutual repulsions of the atoms of the groups of four around the points 

 A wiU preponderate. Hence for some value of r between and a there will be 

 equilibrium. There may, according to the law of force, be more than one value of 

 »• between and a giving equilibrium ; but whatever be the law of force, there is 

 a value of r giving stahle equilibrium, supposing the atoms to be constrained to the 

 lines OA, and the distances r to be constrainedly equal. It is clear from the sym- 

 metries 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 homogeneous equilateral distribution of 8-atom groups, in equilibrium. 

 Similarly, by placing atoms on the three diagonals BjOB^, B.^OBj, BgOBg, through 

 the six tetrahedral angles of the dodekahedron around O, we find a homogeneous 

 equilateral distribution of 6-atom groups, in equilibrium. 



Place now an atom at each point 0. The equilibrium will be disturbed in 

 each case, but there will be equilibrium with a different value of r (stiU between 

 and «). Thus we have 9-atom groups and 7-atom groups. 



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 distri- 

 bution of 3-atom groups in equilibrium by placing an atom at every point O, and at 

 each of the eight points Aj, Aj, A,, Ag, Aj, A., A^, Ag. This we see by observing 

 that each of these eight A's is common to four tetrahedrons of A's, and is in the centre 

 of a tetrahedron of O's ; because it is a common trihedral corner point of four con- 

 tiguous dodekahedrons. 



Lastly, choosing A^, A3, A4, so that the angles AjOAj, AjOAg, AjOA^ are each 

 obtuse,' we make a homogeneous assemblage of 2-atom groups in equilibrium by 

 placing atoms at 0, Aj, A,, A3, A^. There are four obvious ways of seeing this as an 

 assemblage of di-atomic groups, one of which is as follows : — Choose A, and as 

 one pair. Through Aj, A3, A^ draw lines same-wards parallel to AjO, and each 

 equal to AjO. Their ends lie at the centres of neighbouring dodekahedrons, which 

 pair with A^, A3, A^ respectively. 



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

 homogeneous assemblage of double atoms is very important. Remark that every 

 is at the centre of an equilateral tetrahedron of four O's ; and every A is at the centre 

 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 

 neighbours ; and O and three of its twelve nearest neighbours. By aid of an il- 

 lustrative 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 the author had communicated to the Royal Society of 

 Edinburgh was illustrated to Section A. 



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

 geneous assemblage of Boscovich atoms, there are in general two different rigidities, 

 «,Mi, and one bulk-modulus, k ; between which there is essentially the relation 



3A; = 3« + 2wi, 



whatever be the law of force. The law of force may be so adjusted as to make 

 Mj = « ; and in this case we have 3^ = 5w, which is Poisson's relation. But no 

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

 blage of doable, or triple, or multiple Boscovich atoms. On the contrary, any 

 arbitiarily chosen values may be given to the bulk-modulus and to the rigidity, by 

 proper adjustment of the law of force, even though we take nothing more complex 

 than the homogeneous assemblage of double Boscovich atoms above described. 



The most interesting and important part of the subject, the kinetic, was, for 

 want of time, but slightly touched in the communication to Section A. The author 

 hopes t.. Huter on it more fully in a future communication to the Royal Society of 

 Edinburgh. 



* This also makes AjOA,, AjOA,, and A,OAf each obtuse. Each of these six 

 obtuse aiigles is equal to 180° — cos*'(i). 



