614 Scientific Proceedings, Royal Dublin Society. 
5. A group containing eight similar balls similarly placed 
will, if the situations of the eight balls are identical, belong to one 
of the classes 21, 22, 28, or 29. If the similar situations of the 
balls form two sets enantiomorphously related it will belong to 
one of the classes 6, 23, 24, 26, or 31. In all cases, except when 
the group belongs to class 6, 22, 23, 24, or 26, the situations of 
the ball-centres will be singular points. Groups of classes 22 or 
29 are not identical with their own mirror-images, consequently 
exist in two enantiomorphous forms. 
The number of groups obtainable by twofold substitution given 
under two heads A and B as before, is as under:— 
NuMBER OF GROUPS DERIVABLE. 
Description of group of similar similarly A B 
situated ball-centres before ; 
substitution is made. 
Balls sub-)| No. of ||Balls sub-| No. of 
stituted | Enantio- || stituted | Enantio- 
alike. morphs. || different. | morphs. 
Hight-Ball Groups. 
8a, Class 21, the eight similar ball-centres 
lie at the corners of a right pee 
prism, 6 1 pair 7 2 pairs 
84, also Classes ‘21, on the four sides of a 
square equidistant from the angles, . 6 — 7 = 
8c, Class 22, form an 8-point-group without 
plane: s of symmetry (an enantiomorph) 6 — Z = 
ae Class 28, at the corners of a cube, 3 — 3 == 
, Class 29, at the corners of a cube (an 
enantiomorph), 3 — 3 — 
8f, Class €, at the corners of a rectangular 
parallelopiped, c 10 3 pairs 14 7 pairs 
89, Class 23, at the corners of a square prism, ‘ 8 3 pairs 14 7 pairs 
8h, Class 24 (for form see Sohncke’s fig.), 10 4 pairs 14 7 pairs 
Si, Class 26, on the four sides of a square ; 
equidistant from the angles, : 8 2 pairs 14 7 pairs 
87, Class 31, at the corners of a cube, 6 4 1 pair 6 3 pairs 
6. A group containing twelve similar balls similarly placed 
wili, if the situations of the twelve balls are édentical, belong to 
one of the classes 9, 10, 28, 29, 30, 31, or 382. If the similar 
situations of the balls form two sets enantiomorphously related, it will 
belong to one of the classes 11, 12, 13, or 17. The situations of 
the ball-centres in groups of classes 9, 28, 29, 30, and 31 are 
