130 



such systems can be generally deduced by the method of Boldyrew 

 and others. The systematical deduction of all possible symmetrical 

 arrangements, being a purely mathematical and very extensive 

 problem, may also, therefore, be omitted here, and only some general 

 properties of these systems be elucidated by suitably chosen examples. 

 Most of the sixty- five Sohnckian systems can be imagined to 

 be deduced from the space-lattices of Bravais by replacing each 

 point of them by definite, similarly composed, groups of points, the 

 symmetry-elements of which are, however, differently oriented with 

 respect to those of the fundamental B 



space-lattice. In fig. 114. a section A c ^ 



through such a system is reproduced, B B 



the points of it being replaced by B C c^ B 



absolutely unsymmetrical repeats. * A ^c* A * 

 The existence of an infinite number * ^ B A 



of tetragonal axes and of an infinite B c $ 



number of binary axes situated A c * B 



between them, and arranged in the * 



^P ~ D ^^ A 



same disposition, is clearly exhibited . 1115 



by the pattern, and also the existence 



of centrical symmetry. Moreover, the characteristic translations 

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

 Something analogous occurs in the symmetrical system, a section 

 of which is represented in fig. 115. 



Here a set of parallel senary axes A is present, while trigonal 

 axes B and digonal axes C alternate with them, in accordance with 



Euler's theorem. If the hexagonal cells 



O O O O O O ^ e reduced to a single point, there results 



O O O O O O an arrangement which is not different 



O O O O O O from the hexagonal space-lattice of Bra- 



O OO OO O vais; but when the hexagons extend 



Q Q Q ^ Q^ ~ ' and reach their neighbours, the result 



O O O O O O w ^ ke an arrangement, a section of 



O O O O O O which is reproduced in fig. 116, and 



Fig. 116. 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 of n congruent and parallel, interpenetrating space- 

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

 same space-lattice are all parallel to each other; they are, however, 



