BarLtow—A Mechanical Cause of Homogeneity of Crystals. 553 
similarly placed in the nearest surrounding cells. Then, as in the 
last example, within the cavities which exist about the centres of 
the cells, insert octahedrally-arranged grouplets, containing six 
spheres each, of such smaller magnitude that they touch the spheres 
first placed. Hach of these smaller spheres will then be in contact 
with four of the same set and four of the larger set, while each of 
the larger spheres will be in contact with two of the smaller set, 
and eight of the larger set. 
The packing of such an assemblage is very close, but if we 
compare the shape of a single compound grouplet with the shape 
of one of the cells, it is easy to see that there is a slight openness 
of the packing at the square interfaces of the cells, that there is a 
hollow space between the four spheres lying on the one side of 
such a face and the four spheres lying on the other side. If now 
at each square interface the centres of two diagonally-placed 
spheres on one side and those of the two spheres opposite to them 
are symmetrically moved slightly towards one another, all four 
centres continuing in the same plane of symmetry, the hollow 
places in the arrangement can be contracted and a possible equili- 
brium arrangement? for balls of two kinds can in this way be 
obtained, in which the centres referred to form a series of similar 
twelve-point groups, which have planes of symmetry parallel to 
the cube faces. 
The type of the arrangement thus indicated is that marked 
‘10a; in my list.” The generic symmetry is that of class 31 of 
Sohncke.* The two kinds of centres are present in the numerical 
proportions 1:2; each forms a singular point system, the less 
numerous centres lying on digonal axes and in planes of sym- 
metry and the more numerous in planes of symmetry. 
Case of holohedrism of hexagonal system. 
If a stack of equal spheres is made, consisting of plane layers 
triangularly arranged in contact, placed so that the spheres of the 
different layers are vertically over one another, a number of 
similar interstices are left between the spheres. Place now in 
these interstices smaller spheres whose radius is such that they 
1 But see Note 5, p.551. ? Zeitschr. fiir Kryst., 23, p.45. 9 Lbid., 20, p. 467 
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