Bartow—A Wechanical Cause of Homogeneity of Crystals. 611 
3. A group containing four similar balls similarly placed will, 
if the situations of the four balls are identical, belong to one of the 
classes 6, 7, 21, 22, 23, 24, 26, 27, 80, or 82. If the similar 
situations of the four ball-centres form two sets of two points 
enantiomorphously related, it will belong to one of the classes 3, 8, 
24, or 25. In all cases, except when the group belongs to class 8, 
7, 8, 25 or 27, the situations of the ball-centres will be singular 
points. Groups of classes 7, 22, 27 or 32 are not identical with 
their own mirror-images, consequently exist in two enantiomor- 
phous forms. 
The number of different groups which can be derived by 
twofold substitution, z.e., by substituting two balls or complexes 
for some two of the four similar balls found in a group, is given 
below in two columns A and B, A giving the number obtainable 
when both substituted balls are alike, B the number when they 
are different from one another. If any of the derived groups are 
enantiomorphs the number of these is given. Those which do not 
pair as enantiomorphs are identical with their own mirror-images, 
except in cases where the original group is itself an enantiomorph. 
NuMBER OF GROUPS DERIVABLE. 
Description of group of similar similarly in B 
situated ball-centres before : ; 
substitution is made. 
Balls sub-| No. of ||Balls sub-| No. of 
stituted | Enantio- || stituted | Enantio- 
alike. morphs. || different. | morphs. 
Four-Ball Groups. 
4a, Class 6, the four similar ball-centres 
lie at the angles of a rectangle, ; 3 — 3 — 
46, Class 7, at either set of the alternate 
corners of a ard gay pee US 
(an enantiomorph), 3 _ 3 — 
4c, Class 21, at the angles of a square, ‘ 2 — 2 — 
4d, Class 22, at the ate of a ere le 
enantiomorph), ; 2 — 2 — 
4e, Class 23, at the angles of a ie a 
enantiomorph), 6 2 _— 3 —_— 
