ON THE STRUCTURE OF CRYSTALS. 



315 



Further, it must be noted that the system of coincidence-movements 

 of fig. 5 does not necessarily possess any planes of symmetry. The mere 

 Sohncke-system of points or a system of spheres placed at the points of 

 tig. 5 would possess planes of symmetry, but a parallel system of un- 

 symmetrical pear-shaped bodies would not. 



The application of Bravais' method to a system of this kind is incon- 

 venient because it is impossible to partition it into identical same-way 

 orientated units of any kind uithouf lon-ering the symmetry by tJie act of 

 partitioning. Thus in the case in question an hexagonal axis is impos- 

 sible for the unit because the hexagonal axes present in the system are 

 none of them mere axes of rotation, and, therefore, the movements about 

 them are incapable of bringing any conceivable unit to coincidence with 

 itself. This renders some important conclusions of Bravais inapplicable 

 to such a system. Thus he argues that in all holohedral crystals the 

 molecular polyhedra possess the same axes and planes of symmetry as 

 the assemblage. Now the system of hexagonal symmetry just described 

 becomes holohedral if it consists of points or spheres lying on planes 



Fig. G. 



drawn through the nearest hexagonal axes, and yet, as just remarked, no 

 kind of partitioning can produce in it units having hexagonal axes. 



Regarded as an investigation of the total number of ways in which 

 identical repetition can take place, and, therefore, as an investigation of 

 the number of types of homogeneous structure so obtainable, Sohncke's 

 work is exhaustive and complete. He begins without any assumption 

 involving knowledge of previous views or methods, and rigidly deduces 

 the total number of types just mentioned. ^ His method, however, is not 

 free from objection, since, in order to account for the thirty-two different 

 classes, he is, like Bravais, driven to make the symmetry of a system 

 depend partly on the arrangement of the ultimate parts or atoms and 

 partly on the configuration of these atoms. He ti'eats the parts repeated 



' See ib., p. iv. : ' . . . ich die ganze Untersnchnng, soweit sie auf Krystallographie 

 Bezug hat, selbststiindig von vorn anfing, natiirlich mit Benutzung des bewiihrten 

 Grundgedankens der Jordan'schen Metbode.' 



