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Bartow—A Mechanical Cause of Homogeneity of Crystals. 547 
attained by massing the two sizes separately in the triangular 
arrangement. 
Let us now pass to the consideration of cases of balls of two 
kinds not confined to the same plane, and all independent of one 
another. 
Suppose that the relation between the sizes of the two kinds is 
such that, when closest-packing is attained in a combination of the 
two, the centres of the large balls are found at the centres of a sym- 
metrically selected half of the cubes of a cubic partitioning of space, 
i.., having the arrangement which would be the closest-packed 
possible if they were present alone'—in other words, that the 
smaller balls are small enough to go into the interstices between 
the larger ones when the latter have the closest-packed ar- 
rangement referred to. It is then 
evident that the smaller balls are 
moperative ones—that they have no 
share in the production of thegeneral 
symmetry, but merely lie loosely be- 
tween the balls whose interaction de- 
termines it, and which therefore may 
be designated operative. 
The larger of the interstices are 
~ equalin number to the closest-packed 
Fig. 8. spheres; and if, insuch anassemblage 
of spheres, the smaller-sized spheres are too large for the smaller 
interstices, while small enough for the larger, they will occupy 
the latter only. 
In an assemblage of mutually-repellent particles of two kinds 
corresponding to this,’ the relative positions of the particles exer- 
cising the lesser repulsions will be given by the centres of spheres 
just large enough to fit into the large interstices, the arrangement 
produced being that of the sphere-centres in fig. 8. In such an 
arrangement, the centres of one kind occupy the centres of half the 
cubes of a system of cubes into which space is divided, and those 
of the other kind the centres of the remaining half, each half 
system consisting of cubes in contact at their edges only. The 
1 That of the sphere-centres in figs. 1 and 2. 2 See page 529. 
