133 



features of the regular systems are evidently preserved in this new 

 arrangement too, and Sohncke has, for instance, proposed systems 

 of this kind to explain the crystal-structure of complex molecular 

 compounds, like salt-hydrates, etc. Moreover, he was able to give 

 a rational explanation in this way of the occurrence of some tetar- 

 tohedral and hemimorphic crystals, which could not be explained 

 by means of his 

 original, unextended 

 theory. 



Another example 

 of two such inter- 

 penetrating systems 

 built up from two 

 different repeats, is 

 the pattern shown 

 in fig. 119. Here the 

 symmetry of the 

 whole pattern is evi- 

 dently the same as 

 of each of its motifs, 

 these having all the 



same tetragonal 

 symmetry. 



The extended 

 theory of Sohncke 

 can be used succesfully for the explanation of the structure of crys- 

 talline chemical compounds, if it be supposed that the points of 

 all interpenetrating space-lattices of such a system are replaced 

 by one and the same kind of chemical atoms; to this we will draw 

 attention again later on. 



13. However, from the above it may be clearly seen that the 

 application of the theories of Bravais and Sohncke to the pro- 

 blems of crystal-structure, always involves to a certain degree 

 certain suppositions about the special properties of the molecules 

 which take the places of the points in the deduced arrangements. 



From a mathematical viewpoint, however, it is of importance 

 to solve the problem : how to find the total number of such arrange- 

 ments of repeats, that the tridimensional patterns produced may 

 have all the 32 symmetries which are possible for stereometrical 

 regular systems, without it being necessary to make any special 



Fig. 119. 



