127 



spect to those of the fundamental space-lattice. In fig. 112 a 

 section through such a system is reproduced, the points of it being 

 it placed by absolutely unsymmetrical repeats. The existence of an 

 infinite number of tetragonal axes and of an infinite number of 

 binary axes situated between them, and arranged in the same dis- 

 l>Mtion, is clearly exhibited by the pattern, and also the existence 

 rentrical symmetry. Moreover the characteristic translations 

 of the fundamental net-plane of the pattern, are easily recognisable. 



Something analogous occurs in the regular system, a section of 

 /hich is represented in fig. 113. * B 



Here a set of parallel senary A .9 



ces A is present, while trigonal B 9 



ces B and digonal axes C alternate * B C * *C B 



th them, in accordance with A C* A * 



Knier's theorem. If the hexagonal * B A 



cells be reduced to a single point, B C * R * 

 there results an arrangement which A c t B 



not different from the hexagonal % ^ 



space-lattice of Bravais; but when 

 the hexagons extend and reach g> II3> 



their neighbours, the result will be an arrangement, a section of 

 which .is reproduced in fig. 114, and which is evidently not met 

 with amongst the regular systems deduced by Bravais. 



Generally speaking, the Sohnckian systems can be considered 

 to be built up from n congruent and parallel interpenetrating space- 

 lattices of Bravais. The repeats placed in the various points of 



the same space-lattice 'are all parallel 



N Q ft JP ft ft to each other; they are however not 

 O O O O O O similarly oriented in the different com- 



O O O O O O posing space-lattices, but they can be 

 O O O O O O brought to successive coincidence with 

 ft Q ft O O O each other by the characteristic motions 



O O O O O O ^ ^ e re u l ar system under consideration. 



O O O O O O An observer placed in the consecutive 



F - 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. 112, if he looks every time in the direction 

 of a quaternary axis of each tetrade of motifs. 



