CHAPTER V. 



The Limits of the axial periods in Crystalline Matter. - H auy's 

 Law. Crystallographically occurring Symmetry-Axes. Groups 

 and Sub-groups; their relation to Holohedral, Hemihedral and 

 Tetartohedral Crystal-Classes. Crystal-systems. Gadolin's 

 Projection of Symmetry-Elements. - The Symmetry-Classes 

 of the Cubic System. - General and special simple Forms. 

 - Symmetry-groups with Axes of Isotropy. - - The Symmetry 

 of a Physical Phenomemon, of a Physical State, and of a Physical 

 Medium. - - The "image" of a Physical Phenomenon. - - The 

 Symmetry of Cause and Effect, and their mutual Relation. - 

 The Symmetry of the Electrostatic and that of the Mag- 

 netic Field. - - The Symmetry of centrically-symmetrical Phe- 

 nomena in Crystals. The Superposition of Different Causes. - 

 Symmetry and Dissymmetry. General Remarks on the symme- 

 trical Arrangement of experimentally determined Numbers. - 

 Problems and Investigations of the Future. - 



1. In the preceding chapters we extended our researches to 

 include all kinds of symmetrical systems. It need hardly be remarked 

 that, as evidently no special circumstances prohibit the occurrence 

 of every kind of symmetry-axes in the objects of living nature, 

 such a general way of treating the problem was the indicated and 

 only effectual one for the application of the doctrine of symmetry 

 in the whole domain of natural science. However in the case of 

 other, non-living natural objects, experience teaches us that by no 

 means such an unlimited variety in the periods of the symmetry-axes 

 manifests itself; and more particularly in the domain of crystalline 

 matter, there must be some reason why certain limits are apparently 

 set to the possible values of the numbers n, and to the characteristic 

 periods of the symmetry-axes, as these are determined by . 



Indeed, in no field of physical research does the significance of 



