Bartow—A Mechanical Cause of H omogencity of Crystals. 616 
singular points. Groups of classes 10, 29, or 32 are not identical 
with their own mirror-images, consequently exist in two enantio- 
morphous forms. 
The number of groups obtainable by twofold substitution, 
given under two heads A and B as before, is as under :— 
Description of group of similar similarly 
situated ball-centres before 
substitution is made. 
Twelve-Ball Groups. 
12a, Class 9, the ball-centres lie at the 
corners of a regular hexagonal prism, 
120, also Class 9, two and two on the sides 
of a regular hexagon cauidisin from 
the angles, : 
12c, Class 10, form a 12-point- group of the 
hexag onal system which has no planes 
of symmetry (an enantiomorph), 
12d, Class 28, at the middle points of the 12 
edges ofa cube, 2 
12e, Class 29, at the middle points of the 
12 edges of a cube (an enantiomorph), 
12f, Class 30, form a specialized regular 12- 
point-group whose points lie in planes 
which pass seuss opposite edges of 
acube, . 
129, Class 31, form a " specialized 12- -point- 
group whose points lie in planes 
through principal axes, . 
12h, form such a 12-point-group further 
specialized so that each point is 
equidistant from 5 nearest points, 
which consequently lie at the angles 
of aregular pentagon; other balls if 
present oe arranged in this higher 
symmetry, + : 
122, Class 32, form a 12- -point- eroup “with- 
out planes of pomeey (an enantio- 
morph), . 
127, Class 11, at the corners of a regular 
hexagonal prism, . 
12%, Class 12 (for form see Sohncke’s s fig.), 
127, Class 13 (for form see Sohncke’s fig.), 
12m, Class 17, on the sides of a regular 
hexagon equidistant from the angles, 
NumsBer oF GROUPS DERIVABLE. 
A 
Balls sub-] No. of 
stituted | Enantio- |} 
alike. morphs. 
9 2 pairs 
9 au 
9 = 
5 1 pair 
5 = 
7 2 pairs 
7 2 pairs 
3 pes 
7 — 
12 5 pairs || 
14 5 pairs 
14 5 pairs 
12 3 pairs 
11 
11 
11 
'Balls sub-| No. of 
stituted | Enantio- 
different. 
B 
morphs. 
4 pairs | 
11 pairs 
11 pairs 
11 pairs 
11 pairs 
1 Although a group of this kind can form the unit of a homogeneous assemblages, 
the symmetry of the group is not such as can be possessed by the assemblage. 
