328 TRANSACTIONS OF SECTION B. 
molecule of which consists of but one atom and in which all the atoms are 
similar. Consideration of this kind of case shows that the set of identically 
similar centres of attracting and opposing forces will be in equilibrium when 
one particular simple condition is fulfilled; the condition is that, with a given 
density of packing of the centres, the distance separating nearest centres is a 
maximum. Two homogeneous arrangements of points fulfil this. condition, and 
these exhibit the symmetry of the cubic and the hexagonal crystalline systems. 
Since the nature of the two arrangements of points is not easily realised by 
mere inspection, the systems must be presented in some alternative form for 
the purpose of more clearly demonstrating their properties; this is done con- 
veniently by imagining each point in either arrangement to swell as a sphere 
until contact is made with the neighbouring points. The two arrangements then 
become those shown in Figs. 1 and 2, and are distinguished as the cubic and 
the hexagonal closest-packed assemblages of equal spheres; they differ from 
all other homogeneous arrangements in presenting maximum closeness of packing 
of the component spheres. The equilibrium condition previously remarked— 
that, with a given density of distribution of the force centres in space, the 
distance separating nearest centres is a maximum——is revealed in the assemblages 
of spheres as the condition that the spheres are arranged with the maximum 
closeness of packing. 
A further step is yet necessary. Each point in the arrangements considered 
is regarded as the mean centre of an atom of the crystalline element, but the 
assumption originally made states nothing about the magnitude of the atom 
itself ; it is therefore convenient to regard the whole of the available space as 
filled by the atoms, without interstices. This is conveniently done by imagining 
tangent planes drawn at each contact of sphere with sphere, so partitioning 
the available space into plane-sided polyhedra, each of which may be described 
as the domain of one component atom. The twelve-sided polyhedra thus 
derived from the cubic and the hexagonal assemblages represent the solid areas 
throughout which each atom exercises a predominant influence in establishing 
the equilibrium arrangement. 
The two assemblages can now be described in a quantitative manner by 
stating the symmetry and also the relative dimensions of each. The cubic 
assemblage exhibits symmetry identical with that of the cube or the regular 
octahedron, a symmetry characteristic of so-called holohedral cubic crystals; 
the relative dimensions in different directions are defined by the symmetry. 
The assemblage can, in fact, be referred to three axes parallel to the edges of a 
cube, and as these directions are obviously similar in a cube, their ratios are of the 
form, a:6:c=1:1:1. This expression indicates that if the assemblage, sup- 
posed indefinitely extended through space, is moved by a unit distance in either 
of the three rectangular directions a, 6, and c, the effect, as examined from any 
point, is as if the assemblage had not been moved at all. 
The symmetry of the hexagonal assemblage is identical with that of a 
hexagonal prism or of a double hexagonal pyramid, and is that characteristic 
of the so-called holohedral, hexagonal, crystalline system; the relative dimen- 
sions are no longer defined entirely by the symmetry, and are conveniently stated 
as the ratio of the diameter, a, of the prism or pyramid, to the height, c¢, of 
the pyramid. The ratio, a:c, for the assemblage of spheres under discussion 
can be calculated ; it assumes two forms, corresponding to two modes of selecting 
alternative principal diameters of the prism as unit. The alternative ratios are: 
a: c=1:1°6330 or a: c=1: 1°4142. 
This somewhat lengthy theoretical discussion has now reached a stage at 
which it can be applied to the observed facts; the accompanying table (Table I.) 
states the mode in which crystalline substances of different degrees of molecular 
complexity distribute themselves amongst the various crystal systems. Of the 
elements which have been crystallographically examined, 50 per cent. are cubic, 
whilst a further 35 per cent. are hexagonal : and consideration of the data for 
these latter shows that they exhibit approximately the axial ratios characteristic of 
the hexagonal closest-packed assemblage; thus magnesium shows a : c=1: 1:6242, 
and arsenic the ratio a: c=1: 1°4025. 
