90 



considered as the limiting cases to which the endospherical groups, 

 the dihedron-groups D n , and the cyclic groups C n , -- both of the 

 first and of the second order, are approaching, when the number 

 n gradually increases. 



On closer examination all endospherical groups appear to approach 

 then to two definite groups, which will be called spherical groups, and 

 to which we shall attribute the symbols Kg and K^ respectively. 



The group Kg is characterised by the possession of an infinite 

 number of axes of isotropy, of an infinite 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 Kg 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 whatsoever for any direction is manifested, 

 the directional qualities thus becoming effaced by this, the 

 system as a whole will yet have the symmetry Kg . 



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 Kg . 



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

 liquid be represented by a small portion of a screwthread, 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 : Dg , Cg , D^ , C , 

 and C^ respectively. 



The group Dg 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. 



