616 Scientific Proceedings, Royal Dublin Society. q 
7. A single type of group containing sixteen similarly-placed q 
balls, in which the similar situations of the balls form two sets — 
enantiomorphously related, belongs to class 21, the situations of 
the balls not being singular points, and the form being identical 
with its own mirror-image. 
NuMBER OF GROUPS DERIVABLE. 
A. B. 
Balls substituted No. of Balls substituted No. of 
alike. | Enantiomorphs. different. Enantiomorphs. 
20 | _ 7 pairs 30 15 pairs 
8. A single type of group containing twenty similarly-placed 
balls can be obtained by occupying the twenty similarly-situated 
points lying midway between every nearest three points of the 
specialized 12-point-group referred to under 12 h. above,’ the form 
obtained being identical with its own mirror-image. 
NuMBER OF GROUPS DERIVABLE. 
A. B. 
Balls substituted No. of Balls substituted No. of 
alike. Enantiomorphs. different. Enantiomorphs. 
6 1 pair 7 2 pairs 
9. A group containing twenty-four similar balls similarly 
placed will, if the situations of the 24 balls are identical, belong to 
one of the classes 28 or 29. If the similar situations of the balls 
form two sets enantiomorphously related it will belong to one of the 
classes 9, 80, or 81. The situations of the ball-centres in class 
28 are singular points. Groups of class 29 are not identical with 
1 When a homogeneous assemblage is built up of such groups, although the twenty 
balls oceupy similar positions so far as a single group is concerned, they will not all 
occupy similar positions in this assemblage. 
