130 



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 in- 

 volves to a certain 

 degree certain sup- 

 positions about the 

 special properties of 

 the molecules which 

 take the places of 

 the points in the 



deduced arrange- 

 ments. 



From a mathe- 

 matical viewpoint 

 however it is of im- 

 portance to solve 

 the problem : how 

 to find the total 

 number of such ar- 

 rangements of re- 

 peats 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 assumption about the 

 nature of the constituent motifs. As we have seen (9), the solution 

 of this problem involves the supposition of two enantiomorphously 

 related repeats, as soon as there is question of patterns having sym- 

 metry-properties of the second order. 



The mathematical problem just mentioned has been solved by 

 Von Fedorow and by Schoenflies 1 ); and although it would be 

 quite out "of place here to give a full account of these deductions, 

 some general remarks as to the way followed by these authors may 

 be of interest. 



Both authors subdivide the unlimited space into an infinite number 

 of equal or enantiomorphously related, contiguous small volumes 



*) loco cit.; see also: A. Schoenflies, Zeits. f. Kryst. 54. 545. (1915); 

 55. 323. (1916); F. Wallerant, Bull, de la Soc. Miner. 21. 197. (1898). 



Fig 117. 



