131 



not similarly oriented in the different composing space-lattices, 

 but they can be brought to successive coincidence with each other 

 by the characteristic motions of the regular system under consi- 

 deration. An observer placed in the consecutive non-parallel motifs 

 of the pattern, will then see the whole infinite system always in 

 the same way, only when he subjects himself to the successive 

 symmetrical operations characteristic of each group of non-parallel 

 motifs; for instance in fig. 114., if he looks every time in the direction 

 of a quaternary axis of each tetrade of motifs. 



In fig. ii j two non-superposable regular systems are reproduced 

 in projection, which are characterised by a set of parallel trigonal 

 screw-axes perpendicular to the plane of the figures; their points are 



9 







9 













d. 



> . 



@ . 







Fig. 117. 



substituted by perfectly asymmetrical repeats. The repeats of three 

 consecutive layers are distinguished by their colour, and they are 

 tinted more darkly, the nearer they are to the observer's eye. It is 

 obvious that we have here two arrangements, characterised by right- 

 and left-handed screw-axes, and being real non-superposable mirror- 

 images of each other. Crystals whose unsymmetrical molecules were 

 placed in the points of these regular systems, would evidently 

 exhibit true enantiomorphism, as, for instance, is often observed in the 

 case of crystalline substances endowed with optical rotatory power. 

 11. With respect to the symmetry of Sohncke's sixty-five 

 regular systems, we may remark here that they are all characterised 

 by rotations and translations, and that their symmetry is exactly 

 the same as that of the symmetry-groups previously deduced, 

 possessing only symmetry-properties of the first order. If the points 

 in these regular systems be substituted by absolutely arbitrary 



