MOLECULES IN A CRYSTAL. 491 



For example, the bees-cell arrangement of the figure may 

 be regarded as two interpenetrating webs with triangu- 

 lar meshes, one formed by the points A, and the other by 

 the points B ; and one of these is the same as the other 

 turned through 6o°. 



Hence a cluster of adjacent points belonging to the 

 different lattices in this new theory corresponds to a 

 molecular cluster forming a particle in the lattice of 

 Bravais. None the less, however, the newer theory is 

 a remarkable advance upon the old, because the sym- 

 metry is deduced in a general way from the arrange- 

 ment of the particles without any auxiliary hypothesis. 



A few years ago, however, Wulff (9) made the curious 

 discovery that one type of symmetry, namely, that character- 

 istic of the rare copper silicate dioptase, is not to be found 

 among the structures of Sohncke. Moreover, among the 

 types of partial symmetry some of the hemimorphic or polar 

 forms, such as those of tartaric acid and struvite, cannot be 

 explained by Sohncke's theory without some assumption 

 regarding the form of the particles themselves ; such an 

 assumption was made by Sohncke, and is just as much an 

 objection to his theory as the corresponding assumption 

 made by Bravais to account for all the merohedral types. 



It, therefore, became necessary for Sohncke to remodel 

 his system so as to include the missing type, and he did so 

 by an assumption which also explains the hemimorphic forms 

 without any auxiliary hypothesis. This principle, which 

 had been previously employed by Barlow (10), consists in 

 supposing two or more different sorts of particles to enter 

 into the crystal structure. 



The interpenetrating lattices which constitute a crystal 

 are no longer identical points, or merely the same lattices 

 turned in different directions, but one consists of particles of 

 one sort, and another may consist of particles of a different 

 sort ; this limitation will evidently reduce the symmetry of 

 the composite lattice, and can be made to include all possible 

 varieties of crystals. 



Here, then, we begin to have a suspicion that crystalline 

 symmetry is not a matter of mere geometry, and to recognise 



