490 SCIENCE PROGRESS. 



In the Bravais theory it was assumed that all the 

 particles are parallel, and with this limitation it is true, as 

 was shown by Sohncke, that the lattice is the only possible 

 homogeneous assemblage. But why are the molecules to 

 be necessarily parallel ? Why, for instance, should not a 

 bees-cell structure like the adjoining figure be possible ? 

 This cannot form any part of a lattice, both because the 



particles do not lie at equal distances 

 along straight lines ; and also because 

 particles at A are not parallel to par- 

 ticles at B ; for A has its three nearest 

 points lying horizontally to the left, and 

 both above and below it to the right, 

 whereas B has its three nearest points 

 lying horizontally to the right, and 

 above and below it to the left ; A is in 

 fact identical with B turned through 6o°. 



Sohncke then investigated (8) all the possible homo- 

 geneous assemblages of points, and came to the conclusion 

 that there are sixty-five (the number was first erroneously 

 given as sixty-six) ; further, that these, if arranged by their 

 symmetry, fall into groups which agree with the various 

 modes of symmetry observed in crystals, including almost 

 all the merohedral types. 



Among these sixty-five structures there are a number of 

 remarkable spiral and "alternating arrangements {e.g., that 

 suggested above for quartz) which seem to have no relation 

 to Bravais lattices ; further, all the possible modes of crystal- 

 line symmetry seem to be explained by the arrangement of 

 similar particles identically grouped throughout the mass ; 

 only with the implied possibility that the particles need not 

 be similarly orientated, but may face different ways. 



Sohncke's theory seems then to be free from the arbitrary 

 assumptions of the Bravais theory, and to be totally distinct 

 from it. 



There is, however, a closer connection between the two 

 theories than is at first apparent ; for it can be proved that 

 each of Sohncke's sixty-five structures consists of two or 

 more interpenetrating lattices inserted into one another. 



