80 



is certainly true, as well as the other view, according to which enantio- 

 morphous figures have never a symmetry- cent re. The above mentioned 

 thesis however is not correct, as has been clearly shown in the preceding 

 chapters. Stereometrical figures are different from their mirror-images, 

 and they can therefore occur in two non-superposable forms, only 

 when they do not possess any symmetry-properties of the second order, 

 whatever they may be. Neither the absence of a symmetry-centre, 

 nor that of a symmetry-plane is therefore sufficient to have enantio- 

 morphism as a necessary consequence. This fact already repeatedly 

 mentioned in the preceding chapters, should be kept in mind, especially 

 by authors on chemical subjects, writing about molecular symmetry; 

 in many textbooks on organic chemistry these relations are wrongly 

 treated. We shall have occasion to return to this subject later on, more 

 especially when we come to deal with Pasteur's law. 



The thirty-two symmetry-groups mentioned can now readily be 

 arranged in a more systematic way if we remember the formerly 

 indicated relations existing between mathematical "groups" and 

 "sub-groups" (p. 72). We have seen that if a number of non-equi- 

 valent operations out of a group of them, be taken so that they 

 may be combined to form a new complete group of operations, this 

 new group is called a sub-group of the original one. The number of 

 non-equivalent operations of a sub-group is always an aliquot part 

 of the number of operations present in the original group. 



Thus, for instance, the group K contains all operations of the 

 group T (p. 72), and therefore T is a sub-group of K. Now while 

 K includes twenty-four non-equivalent operations of the first order, 

 T has just half that number, i. e. twelve', etc. 



In crystallography it is usual to reunite all sub-groups g lt g 2 , 

 g 3 , etc., of another higher symmetrical group G Q , with that group 

 GQ, and form them together into one and the same crystal-system. 



Because of the fact that the number of non-equivalent operations 

 of these sub-groups is always an aliquot part of that of the principal 

 group, and that therefore this is also the case with the number of 

 the limiting faces of the crystals, if they are bordered by the most 

 unrestricted simple form of every class, - - these sub-groups are 

 distinguished from the principal one, by the names hemihedral and 

 tetartohedral groups respectively, while the principal group itself 

 is called the holohedral group. 



This gathering of the sub-groups with their principal one into 

 a crystal-system, has many practical advantages. One of the most 

 important being that all crystal-forms belonging to the same crys- 

 tal-system, can be described with respect to the same set of coordinate- 





