ON THE STRUCTURE OF CRYSTALS. 



327 



these cells the remainder can be found by means of the group of opera- 

 tions. Most of the 230 types can be partitioned into space units which 

 individually possess the symmetry of the system as a whole. When this 

 is the case, a finite group of contiguous elementary cells will form such a 

 unit and can be found by applying to one of them certain of the elements 

 of symmetry which lie on its surface ; a symmetrical space unit of this 

 kind may be called a complex cell.'^ Other complex cells possessed of less 

 symmetry can, of course, be formed. Some of the 230 types, while 

 capable of being partitioned into such less symmetrical complex cells 

 cannot be partitioned into complex cells which have as high a symmetry 

 as that of the type. For example, the type represented by the Sohncke 

 system described above (fig. 5, p. 314) can be partitioned into cells pos- 

 sessing trigonal axes with or without centres of symmetry or planes 

 of symmetry, or with both, but its cells cannot individually possess an 

 hexagonal axis. 



As an example, take the case of the hexagonal space-lattice of 



Fig. 9. 



£g. 2, where the axes are axes of rotation and the planes of symmetry 

 are planes of reflexion. The shape of the bodies placed at the points is 

 ignored, or in other words they are supposed to have a symmetry which 

 does not modify that of the system of arrangement. The points H in 

 fig. 9 constitute such a space lattice. In this figure H2H4H6HJ;, corre- 

 spond respectively to a, y, h, fy of fig. 6. H,H2 is an hexagonal axis of rota- 

 tion. Take H; as the origin of this Bravais-system, which we know is 

 a perpendicular prism with a rhomb of 60° as base. All the rows of 

 the system parallel to H,H.2 are also hexagonal axes of rotation. By 

 combining these rotations with the translations of the system we see at 

 once that straight lines such as TjTa, T3T4, which are parallel to the 

 hexagonal axes and pass through the centres of gravity of the equi- 

 lateral triangles forming the bases of the lattice, are trigonal axes of 

 rotation; and, again, straight lines such as D1D2, D3D4, D-jI^r,) D7DS, 

 DgDiQ, which pass through the middle points of the rows of the base, are 

 digonal axes of rotation. 



J Krystalhy Sterne und Krystallstructitr, p. 576. 



