612 Scientific Proceedings, Royal Dublin Society. 
NumsBer or Grours DERIVABLE. 
Description of group of similar similarly A = 
situated ball-centres before ; : 
substitution is made. an 
Balls sub-| No.of ||/Balls sub-| No. of 
stituted | Enantio- || stituted | Enantio- 
alike. morphs. || different. | morphs. 
Four-Ball Groups—continued. 
4f, Class 24, at the corners of a right 
tetrahedron,+ 3 1 pair 3 1 pair 
4g, Class 26, at the angles of a square 2 — 3 1 pair 
4h, Class 27, at the angles of a anu (an 
enantiomorph), 0 2 — 3 — 
4i, Class 30, at the corners of a regular 
tetrahedron, . < : : 5 1 — 1 _ 
4;, Class 32, at the corners of a regular 
tetrahedron (an enantiomorph), . 1 — 1 — 
4k, Class 3, at the angles of a rectangle, 4 1 pair 6 3 pairs 
41, Class 8, at the angles of a rectangle, 4 1 pair 6 3 pairs 
4m, Class 24, at the angles of a square,” 38 1 pair 4 2 pairs 
4n, Class 25, at the corners of a Ge. 
tetrahedron,? 2 4 1 pair 6 3 pairs 
4. A group containing six similar balls similarly placed will, 
if the situations of the 6 balls are identical, belong to one of the 
classes 9, 10, 11, 12, 18, 15, 17, 18, 28, 29, 30, 31 or 32 in 
Sohncke’s list. If the similar situations of the six balls form two 
sets enantiomorphously related, it will belong to one of the classes 
12, 14,16, or 19. In all cases, except where the group belongs to 
class 14, 15, 16, 18, or 19, the situations of the centres will be 
singular points. Groups of classes 10, 15, 18, 29, or 32 are not 
identical with their own mirror-images, consequently exist in two 
enantiomorphous forms. 
1 The ball-centres lie in the planes of symmetry. The tetrahedron is not regular. 
2 The ball-centres lie in digonal axes. 
3 Not a regular tetrahedron. 
