610 Scientific Proceedings, Royal Dublin Society. 
singular points,! 7. e., lying in axes, or in planes of symmetry, or 
in both of these, except in the case of class 5. Groups belonging 
to classes 5, 7, 10, 15 or 22 are not identical with their own 
mirror-images, and consequently existin two enantiomophous forms. 
If the situations of the two balls are enantiomorphously 
similar they may be found in any group whose symmetry is 
that of either of the classes 1, 3, 4, 11, 14, 16, 23, or 25. 
Twotold: substitution in a group of either of these two sets, 
if the two balls or complexes substituted are both of the same 
kind, can, it is evident, be made in one way only. And if the 
two balls substituted are of two different kinds, and the situations 
of the original balls are identical, such substitution can still be made 
in only one way. 
Tf, however, the situations of the original balls are only enantio- 
morphously similar, as in the classes last mentioned, and the two 
balls substituted are of two different kinds, ¢wo different groups are 
obtainable from the original group, and these are enantiomorphs. 
2. If a group contains ¢hree similar balls similarly placed, and 
no others occupying similar situations, it is evident that they 
must lie at the angles of an equilateral triangle, and that the 
number being odd, the existence of enantiomorphously similar 
situations for any of the three balls is precluded. 
And three identical situations, without other similar ones, can 
be found in groups of either one of the classes 13, 15, 16, 19, or 
20 in Sohncke’s list, the required positions for the centres being 
singular points except in the case of class 20. Groups belonging 
to classes 15 or 20 are not identical with their own mirror-images, 
consequently exist in two enantiomorphous forms. 
T'wofold substitution in any such group containing three balls 
can be carried out in one way only when the substituted balls are 
both of the same kind. When they are of two different kinds 
the same is true in the cases of groups of classes 13 or 15, but two 
different groups are obtainable where the original group is of 
class 16, class 19 or class 20, and these two derived groups are 
enantiomorphs when the group from which they are derived belongs 
to class 19. 
1 Compare Zeitschr. f. Kryst., 23, p. 60. 
