1888 - 89 .] Sir W. Thomson on Constitution of Matter. 699 
examples of homogeneous assemblages of symmetrical groups of 
points, with the condition of equilibrium for each when the mutual 
forces act. 
§21. Preliminary . — Consider an equilateral* homogeneous assem- 
blage of single points, 0, O', &c. Bisect every line between nearest 
neighbours by a plane perpendicular to it. These planes divide 
space into rhombic dodekahedrons. Let AjOA^ A 2 OA 6 , A 3 OA Y , 
A 4 OA 8 , be the diagonals through the eight trihedral angles of the 
dodekahedron inclosing O, and let 2 a be the length of each. Place 
atoms Q ls Q 5 , Q 2 , Q 6 , Q 3 , Q 7 , Q 4 , Q 8 , on these lines, at equal dis- 
tances, r, from 0 ; and do likewise for every other point, O', O", 
&c., of the infinite homogeneous assemblage. We thus have, 
around 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 a - r. 
§ 22. Suppose, now, r to be very small. Mutual repulsions of the 
atoms of the groups of eight around the points O will preponderate. 
But suppose 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 zero and a, there will be equilibrium. 
There may, according to the law of force, be more than one value of 
r between zero 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 constrained to the lines 0 A, and the distances r to 
be constrainedly equal. It is clear from 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 homogeneous equilateral distribution of 
8-atom groups in equilibrium. Similarly, by placing atoms on the 
three diagonals, B x OB 4 , B 2 OB 5 , B 3 OB 6 , through the six tetrahedral 
angles of the dodekahedron around O, we find a homogeneous equi- 
lateral distribution of 6-atom groups in equilibrium. 
§ 23. Place, now, an atom at each point O. The equilibrium will 
be disturbed in each case, but there will be equilibrium with a 
* This means such an assemblage as that of the centres of equal globes piled 
homogeneously, as in the ordinary triangular-based, or square-based, or oblong- 
rectangle-based, pyramids of round shot, or of billiard-balls. 
