Bartow—A Mechanical Cause of Homogeneity of Crystals. 609 
Now the variety of types of grouping possible in which some 
given number of similar balls can be grouped about a centre in 
similar situations in such a manner that the group is competent to 
form part of a homogeneous assemblage, is not very large, and the 
number of different ways in which a certain number of similar balls 
of any particular group of this nature can be exchanged for other 
balls or complexes differing from them is easily ascertainable. 
The following is a short enumeration of all the types of group- 
ing of similar balls possible, and it is accompanied by a statement 
of the number of different types obtainable by twofold! substitution, 
7.e., of the number of different groups derivable by the exchange of 
two of the original similar similarly-placed balls for other balls or 
complexes different from them. In enumerating the groupings 
not only typesin which the similar balls occupy identical situations 
are given, but also those in which the situations are of two kinds 
enantiomorphously similar. The number of groups derivable in 
any case by the substitution of different balls or complexes depends 
not only on the arrangement of the balls some of which are 
removed, but also on the arrangement of other balls forming part 
of the same group, if any such are present.” 
It will be convenient to refer to the diagrams in Sohncke’s list 
of Krystallklassen contained in Zeitschr. fir Kryst., xx., p. 457. 
In every case in which the similar balls occupy identical 
situations in the group, it is manifest that, whatever the number or 
arrangement of these balls, substitution for one only will produce 
identically the same effect whichever of the similar balls is selected 
for removal. In cases, however, where the situations are of two 
kinds enantiomorphously similar to one another, single substitution 
of the same ball or complex can be made in two ways, resulting in 
two different groups which are enantiomorphs. 
‘1. As to the number of different types of groups existing in 
which two balls are similarly situated, we see that two identical 
situations without other similar ones, can be found in groups of 
either one of the classes 3, 5, 6,7, 8, 9,10, 12, 18, 15, 21, 22, or 24 
in Sohncke’s list just referred to, the required centres being 
1 Threefold, fourfold, &c., substitution are also readily traceable, but the results 
would consume too much space here. 
2 Compare Bischoff’s ‘‘ Handbuch der Stereochemie,’’ § 3, p. 635. 
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