Barrow—A Mechanical Cause of Homogeneity of Crystals. 555 
Case of hemimorphism of rhombohedral symmetry. 
If tetrahedrally arranged grouplets, each composed of four 
equal spheres in contact, be so placed at the centres of the cubes 
of a cubic partitioning of space that the axes of the grouplets 
coincide with the diagonals of the cubes, the grouplets being all 
similarly orientated, and if the spheres are of such magnitude 
that the grouplets touch one another, then it will be found that 
the spheres have the closest-packed arrangement of figs. 1 and 
2. 
Suppose, now, that instead of all the grouplets being composed 
of spheres of the same size, half of them symmetrically placed—i. e., 
placed about the centres of one of the half systems of cubes 
which touch one another at their edges only—are composed of 
rather smaller spheres. The cubic arrangement will not now give 
closest-packing; for if the largest grouplets be placed as in the 
case just referred to, the smaller ones will be just too small to fill 
the spaces allotted to them. Closest-packing for a mixed assem- 
blage' will, however, probably be reached if, keeping the grouplets 
intact, a slight contraction of the assemblage along the direction 
of one of the cube diagonals be made so as to obtain additional 
contacts, still keeping the structure homogeneous.” The type of 
homogeneous structure thus obtained is that marked 51b,; in my 
list,? and it has the generic symmetry of class 19 in Sohncke’s 
list. The two kinds of centres are present in equal numbers. 
It is evident that in an equilibrium arrangement of this nature, 
centres of the same kind do not all occupy similar positions in the 
structure; one out of every four centres of each kind lies on a 
trigonal axis in which planes of symmetry intersect one another, 
and each of the remaining three centres of each group of four lies 
in a single plane of symmetry. While therefore all the centres 
form singular point-systems, two of these systems contain fewer 
points than do the remaining two. 
1 See note 5, p. 551. 2 See note 1, p. 531. 
3 Zeitschr. fiir Kryst., 23, p. 55. 4 Ibid., 20, p. 463. 
