135 



they are always arranged in a space-lattice characterised by a definite 

 group of translations, etc. l ) 



It is easy to demonstrate, moreover, that no existing symmetry- 

 elements can ever lie within the fundamental domain of a regular 

 structure, but that they are always situated on its surface. This 

 follows immediately from the fact that each symmetrical operation 

 must always bring a fundamental domain into coincidence with 

 another one present in the whole complex. From this it is clear that 

 the existence of symmetry-axes and of symmetij--planes in the 

 structure will then of course 

 be in some way determinative 

 for the shape of the funda- 

 mental domain, as e. g. sym- 

 metry-planes must be always 

 limiting parts of the surface 

 of such fundamental cells (fig. 

 120). In the latter cases it also 

 becomes clear that in general 

 to every fundamental domain 

 A, a second one A', being the 

 mirror-image of the first, must 

 be present, because the reflec- 

 tion of the elementary volume 

 A in the symmetry-plane will 

 change it into its contiguous, 

 but in general non-superpo- 

 sable mirror-image A', etc. 



Within all such enantiomorphously related fundamental domains 

 the whole distribution of matter must of course also be enantio- 

 morphous; and this is the meaning of the supposition of Von 

 Fedorow and Schoenflies, when they maintain that crystals 

 possessing symmetry-properties of the second order must be built 

 up by certain atom-complexes (crystal-molecules) which are of two, 

 enantiomorphously related, kinds. Only in the cases of enantiomor- 

 phous crystals the right-handed and left-handed crystals can separate- 

 ly be composed of atom-complexes of one and the same kind, right- or 



Fig. 120. 



1 ) The "fundamental domain" plays the same role here, as what by G. Friedel 

 is called the "mtftif" of the crystalline aggregation. The latter is identical with our 

 "motif", as soon as the empty space round it is included also within the con- 

 siderations about it. 



