j 
Bartow—A Mechanical Cause of Homogeneity of Crystals. 589 
case, the distances between the centres of a group must be but 
little less than the corresponding distances between next centres 
of different groups. The modified assemblage which results will 
have angles slightly different from right angles, and will belong 
_ to type 62a,, so that it will display the symmetry of class 3 in 
Sohncke’s list ; the situations of the smaller balls will be singular 
points lying in planes of symmetry, but not all alike in the 
unbroken assemblage; those of the larger ones will be singular 
points on digonal axes and on planes of symmetry on the two 
sides of which the corresponding points are not directly opposite. 
In all the above cases the situations of all centres of the same 
kind in a group are identical, and consequently the ties which 
bind them are respectively similar and equivalent.” This, however,. 
is evidently not always the case, for we know that balls of the 
same kind may be differently situated in a homogeneous structure, 
and the same may hold true in a group. 
It has been concluded above that there are countless cases in 
which the principle of closest-packing, where balls of different kinds. 
are used, is productive of symmetrical assemblages, some of them 
capable, some incapable of being partitioned into units of a single. 
kind, without lowering the type of symmetry. And when assem- 
blages capable of this symmetrical partitioning are taken, and by 
some change of the external conditions their balls are aggregated 
into groups of a single kind, the balls of a group being linked 
with one another but not with balls of surrounding groups, a 
further application of the same principle will intermix these assem- 
blages with other balls or complexes of balls, that is provided the 
combination produces still closer packing. 
1 Not, however, in all cases identical considered with respect to the unbroken 
assemblage. 
2 Comp. theory of ring-formation in ‘‘ Handbuch der Stereochemie,’’ yon Dr. C. A. 
Bischoff und Dr. Paul Walden, Frankfurt-a-M., 1894, p. 50. 
The symmetry of arrangement of some of the groups here described, and of other 
groups constructed in a similar way, appears to be inconsistent with the diagrammatic 
linking together of atoms in a molecule ordinarily adopted, but it must be remembered 
that the basis of this graphic conception of links lies rather in a relation subsisting 
between the atom and the group in which it occurs than in one between the atom and 
individual atoms round about it. The graphic representations ordinarily employed may 
therefore very possibly express too much, and the method require some modification. 
See some remarks on valency made later, p. 681. 
