048 Scientific Proceedings, Royal Dublin Society. 
structure is therefore of the type marked 6a, in my list.1 Hach 
kind of centre forms a singular point system,’ centres of both 
kinds lying at points of intersection of tetragonal, trigonal, and 
digonal axes, and also in planes of symmetry. The generic sym- 
metry is the holohedral cubic, being that of class 28 in Sohncke’s list.* 
The use of spheres somewhat smaller than those which would 
just fit into the larger interstices would still leave the arrange- 
ment referred to the closest-packed possible, so long as they 
are not small enough for more than one to go into one of 
these interstices. Corresponding to this we have the fact that 
the arrangement described is an equilibrium one for particles of 
two kinds, if when thus arranged the mutual repulsion between 
the particles of different kinds is either equal to or /ess than the 
repulsion between the particles of the same kind which have the 
closest-packed arrangement of figs. 1 and 2; provided the repul- 
sion between the particles which correspond to the smaller spheres, 
and also that between these particles and particles of the other 
kind are considerable enough to prevent more than one particle 
from occupying any of the larger interstices, and any particles at 
all from occupying the smaller ones. 
If spheres of two sizes are used, and the same arrangement of 
the larger spheres prevails, but the smaller spheres are small 
enough to go into the smadler interstices between the larger ones, 
closest-packing will be attained in some arrangement which is 
homogeneous so far as the disposition of the larger spheres is 
concerned if the arrangement of the smaller ones be neglected, 
but unhomogeneous if the smaller ones are taken into account, 
unless indeed the relative magnitude of the two kinds is such that 
the larger interstices are packed fullest when a number of the 
smaller spheres are arranged in each of them in a manner con- 
sistent with the homogeneity of the structure formed.* 
In these and in all other cases of assemblages consisting of 
more than one kind of ball, it is manifest that unless the different 
kinds are present in the numerical proportions in which they enter 
into the particular combination which gives closest-packing of 
1 Zeitschr. fiir Kryst., 23, p. 44. 2 Tbid., p. 60. 
3 [bid., 20, p. 466. 4 See note 1, p. 531. 
Pp 
