; 
_ Bartow—A Mechanical Cause of Homogeneity of Crystals. 585 
The nature of the symmetry when all the balls composing 
a group are of the same kind has already been described for some 
simple cases.1. For cases generally it suffices to say that any homo- 
geneous finite partitioning of a homogeneous assemblage of balls 
will conceivably furnish possible groups. In order that the parti- | 
tioning may not be arbitrarily incomplete, é.c., that wherever it 
separates two balls which have a certain relative situation, it may 
similarly sever the balls of every similarly related set, the parti- 
tioning must, however, have as high a degree of symmetry as that 
possessed by the assemblage partitioned, ¢.e., must be compatible 
with the coincidence movements (Deckbewegungen) of the latter ;? 
it is imperative too that the partition walls do not intersect any 
ball centres. The partitioning may be such as to produce one, 
two, three or more kinds of groups. The symmetry of the groups 
may be higher or lower than the generic symmetry of the assem- 
blage from which they are carved. 
As the breaking of any link is, as we have said, accompanied 
by the breaking of all similar links, it is evident that each ball 
must occupy a situation with respect to the other balls of the same 
group different from that it occupies with respect to the balls of 
other groups. Obedience to this rule is observed in the examples of 
partitioning into growps given below. 
If the distances separating unlinked centres are greater than 
the distances separating linked centres, it is evident that so far as 
these distances remain unaffected by the nature of the grouping, 
the smaller the groups the greater will be the degree of expansion 
of the assemblage. 
Onit Groups. 
Ifa homogeneous assemblage is composed of groups of one 
kind only, and contains no unaggregated balls, it is evident that 
each group must contain balls of all the different kinds present in 
the assemblage in the numerical proportions in which they occur. 
In such a group the kind or kinds of which there are fewest may 
be represented by a single ball, or by two or more balls according 
to the size of the groups. 
1 See pp. 535, 539 e¢ seg. and also p. 556. 
2 Comp. Mineralogical Mag., vol. xi., p. 180; or Zeitschr. fiir Kryst, 27, p. 460. 
