BarLtow—A Mechanical Cause of Homogeneity of Crystals. 561 
arrangement described. A yet closer packing is however, attain- 
able if all the grouplets of the third set experience a slight sym- 
metrical rotation about their centres, the rotation being of such a 
nature that all the four spheres of each of these grouplets continue 
to occupy similar positions in the four cubes which meet in the 
cube edge. 
In the closest-packed system of spheres of three sizes that can 
be arrived at in this way,! it will be found that each cubic cell 
of the space-partitioning contains a similar arrangement of the 
spheres, the sphere centres of the first and second sets lying, in 
each case, at the angles of a regular tetrahedron, and the sphere 
centres of the third set forming a “ 12—punkter” of Sohncke.? 
The type of homogeneous structure is that marked 7 in my list,° 
and it has the generic symmetry of class 82 in Sohncke’s list of 
Krystallklassen. The two sets of less-numerous centres form two 
singular point-systems whose points lie on trigonal axes. ‘he 
three kinds of balls are present in the numerical proportions 
1:1: 3. There are two equilibrium arrangements of the same 
spheres, one of which is the 
mirror image of the other. 
Centres of the same kind 
occupy similar positions in 
these arrangements. 
Case of pyramidal hemihedrism 
of the hexagonal system. 
Arrange a plane layer of 
equal spheres touching one 
another, but with symmetri- 
cally-situated gaps as shown 
in fig. 15, the gaps having a 
triangular arrangement. 
1 There will probably have to be some linking, see note 5, p. 551. 
2 As in some of the previous cases, the centres of the spheres in the closest-packed 
homogeneous assemblage of spheres thus indicated will not precisely give a possible 
equilibrium arrangement for mutually-repellent particles, this beimg precluded by the . 
necessities of statical equilibrium. (Compare note 2, p. 550). 
3 Zeitschr. fiir Kryst., 23, p. 18. 
2T2 
