MOLECULES IN A CRYSTAL. 495 



types may yet be deduced from one single geometrical 

 assumption ? 



The answer to this suggestion is supplied by another 

 theory of crystal structure, which has meanwhile entered 

 the field in a somewhat striking manner. 



THE PURE STRUCTURAL THEORY. 



Independently and almost simultaneously important in- 

 vestigations have been published by Fedorow (17) and 

 Schonflies (18) upon the possible structures of geometrical 

 figures and their symmetry. 



Fedorow's researches were in reality published before 

 those of Schonflies, but being in the Russian language, 

 attracted no attention among mathematicians. The two 

 authors have come to one and the same conclusion ; namely, 

 that there are 230 possible structures for crystals, and that 

 these, if classed by their symmetry, fall into the now well- 

 known thirty-two crystal groups. This absolute coincidence 

 of results proves, if proof were needed, that the problem has 

 been correctly solved ; but before taking it for granted that 

 the problem has been correctly stated, we must see what 

 these authors mean by a crystalline structure, and in what 

 manner their definition differs from that of others. 



Schonflies defines a crystal as consisting of " absolutely 

 similar molecules, so arranged that each molecule is en- 

 vironed in the same way by all the other molecules ". 



Sohncke's definition of a system of points which is to 

 have the characters required by a crystal structure is that 

 " round every point the arrangement of the remainder is 

 the same as round every other point ". 



At first sight these definitions appear to be identical. 

 How is it, then, that Schonflies arrives at 230 possible 

 structures, while Sohncke finds only sixty-five ? The 

 difference, it will be found, lies in their ways of defining 

 a similar environment. 



Without entering into details, we may in a few words 

 contrast the two methods. 



Sohncke takes any point P of his system and joins it to 



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