Bartow—A Mechanical Cause of Homogeneity of Crystals. 617 
their own mirror-images, consequently exist in two enantiomor- 
phous forms. 
The number of groups obtainable by ¢wo-fold substitution given 
under two heads A and B as before, is as under :— 
NuMBER OF GROUPS DERIVABLE. 
Description of group of similar similarly 
situated ball-centres before 
substitution is made. 
A B 
Balls sub-| No. of ||Balls sub-| No. of 
stituted | Enantio- || stituted | Enantio- 
alike. morphs. || different. | morphs. 
Twenty-four-Ball Groups. 
24a, Class 28, the 24-ball-centres arranged 
to form a specialized 24-point-group 
whose points lie in planes drawn 
through the cube centre perpendinuies 
to the three principal axes, 16 4 pairs 23 8 pairs 
245, also Class 28, 24-point-group whose 
points lie in planes drawn through 
opposite cube edges, : c ; 16 5 pairs 23 10 pairs 
24c, Class 29, form a 24-point-group without 
planes of symmetry (an enantiomorph) 16 — 23 — 
24d, Class 9 (for form see Sohncke’s fig.), . 30 11 pairs 46 23 pairs 
24e, Class 30 (for form see Sohncke’s fig.), 26 10 pairs 46 23 pairs 
25e, Clsss 31 (for form see Sohncke’s fig.), 26 11 pairs 46 23 pairs 
10. A group containing forty-eight similar balls, the similar 
situations consisting of two sets enantiomorphously related, can be 
formed of class 28, and the number of groups obtainable by ¢wo- 
fold substitution from such a group is :— 
NuMBER OF GROUPS DERIVABLE. 
A. B. 
Balls substituted No. of Balls substituted No. of 
alike. Enantiomorphs. different. Enantiomorphs. 
56 23pairs 94 | 47 pairs 
In precise harmony with the conclusion stated above, that in 
