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The group K^ is characterised by the possession of an infinite 

 number of axes of isotropy, by that of an. in finite number of symmetry - 

 planes, and by the presence of a symmetry-centre. 



The symmetry of the group K^ consists in the presence of an 

 infinite number of axes of isotropy, but it does not possess any 

 symmetry-planes, nor a centre of symmetry. 



The symmetry of the group K^ is the highest symmetry which 

 can eventually be attributed to a system. Each arrangement of an 

 infinite number of points deprived of all qualities, or which are 

 at least deprived of all "directional" properties, represents a system 

 having this symmetry; and even if directional qualities of a certain 

 kind are present, but the points are distributed in space in such a 

 way that no preference whatever for any direction is manifested, 

 the directional qualities thus becoming effaced by this, -- the 

 system as a whole will yet have the symmetry K^ . 



If however this effacing influence of the distribution in space is not 

 present, then the occurrence of such "directional" qualities will 

 have as a necessary consequence that the symmetry of the system 

 becomes a lower one than that represented by the group K^ . 



Thus, if the physical state of every molecule of an optically active 

 liquid be represented by a small portion of a screw-thread, a 

 sphere filled with such a liquid can be considered as an object 

 having the symmetry of the group K^ , an infinite number of 

 axes of isotropy still being present, but no planes of symmetry, 

 nor a symmetry-centre. 



Furthermore there are five other groups possible which possess 

 a single axis of isotropy A^ ; in connection with their intimate 

 relations to the dihedron- and cyclic groups of the first and second 

 order, we shall denote them by the symbols: Z)^, C^, D^, , 

 and CQQ respectively. 



The group D^ has a single homopolar axis of isotropy A^, a plane 

 of symmetry perpendicular to it, an infinite number of symmetry- 

 planes passing through A^, and a centre of symmetry. 



A cylindrical flask filled with a hypothetical, homogeneous, and 

 weightless liquid, may be mentioned as an instance of a system 

 having the symmetry D^ . 



The group C**, possesses: a homopolar axis of isotropy A^, 

 a plane of symmetry perpendicular to it, and a symmetry-centre, but 

 no planes of symmetry passing through A^ . 



If a cylinder with circular base be rotated round its axis in a 



