BarLtow—A Mechanical Cause of Homogeneity of Crystals. 543 
_ cube centre or a cube angle, where d is the diameter of a cube of 
one of the two space partitionings employed in this system. 
(As in the previous case, if the distance were exactly d/4, the balls 
would have the closest-packed arrangement shown in figs. 1 and 2). 
The structure is now of the type numbered 9a, in my list; the 
generic symmetry is the holohedral cubic; one half of the groups 
have their orientation opposite to that of the other half. The centre 
points of the groups lie, one set at half the cube centres of a cubic 
partitioning of space, the other set, which are oppositely orientated, 
at half the cube angles. 
Groups composed of four similar similarly-placed balls arranged 
in some other way, e. g., at the angles of a square, do not appear to 
be capable of very close-packing in a homogeneous manner when 
taken alone. 
Take next a case in which the centres, while all of one kind, are 
similarly linked to one another to form groups of six. 
Several different kinds of grouping of six centres are possible 
consistently with this;* let us take the simplest case, that in 
which they lie at the angular points of a regular octahedron. 
Suppose— 
(aa.) That the distances between the six ball centres linked 
together to form a group are small as compared with the distances 
between the nearest centres in different groups. 
As in the case of groups of four similar centres, if the distances 
between the centres of the same group are relatively so small that 
_ the shape of the groups does not affect their relative arrangement, 
we shall get closest-packing in an unhomogeneous arrangement 
in which the relative situations of the centre points of the groups 
approximates to that of the sphere-centres of figs. 1 and 2. 
Suppose— 
(6b.) That the distances between the six ball centres forming 
a group are larger than in case aa, so that the form of the groups 
affects their arrangement. 
Closest-packing will probably not now be attained in any 
homogeneous arrangement belonging to the cubic system ; for, if 
the groups, appropriately oriented, be arranged with their centre 
1 See p. 612. 
SCIEN. PROC. R.D.S., VOL. VIII., PART VI. 28 
