Bartow—A Mechanical Cause of Homogeneity of Crystals. 541 
Several different kinds of grouping of four ball centres are 
possible consistently with this; let us take the most symmetrical 
ease, that in which the centres lie at the angular points of a 
regular tetrahedron. 
Suppose— 
(A.) That the distance between two centres of a group is small 
as compared with the distances between nearest centres in different 
groups. ‘ 
As in the above cases in which a relation of this kind obtains, 
elose-packing will be attained when the arrangement of the groups 
is like that of the sphere centres in figs. 1 and 2, and an arrange- 
ment approximating to this would give the closest-packing possible 
in cases where the distance separating centres of the same group is 
relatively very small indeed. 
We see, however, that when, having this arrangement, the 
groups are orientated in such a way that the assemblage shall be a 
homogeneous structure belonging to the cubic system (?.¢., when 
the ball-centres lie upon trigonal axes) closest-packing is not 
attained; and therefore we conclude that in cases where the groups 
approximate to the arrangement referred to, the orientation is less 
regular than this, and the assemblage in consequence either 
unhomogeneous or of a lower degree of symmetry than the 
cubic. 
If the arrangement is homogeneous, but has some lower sym- 
metry than the cubic, the four centres of a group will not all bear 
the same relation to the structure. For example, if the symmetry 
is trigonal, we shall find not more than three of each group occupy- 
ing similar positions, and if three are similarly related the fourth 
will lie on some trigonal axis. 
Suppose next — 
(B.) That the ball centres of the same tetrahedral group are 
further apart than in the case of A, but still much closer together 
_ than centres in different groups. 
It would appear that cases now present themselves in which 
closest-packing is reached when the arrangement of the groups is 
that of the centres and angles taken together of the cubes in a 
1 See p. 611. 
