92 



And because the two operations mentioned are together equivalent 

 to an inversion, the rotating cylinder is evidently congruent with 

 its inverse image, which means that it has itself an inversion-centre. 



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

 and an infinite number of binary' axes perpendicular to it. 



As the group D^ does not possess a symmetry-centre (just as in 

 the case of D n ), the symmetry can be also described by considering 

 A^ as a screw-axis of infinitely small period, with an infinitely 

 small corresponding translation in the direction of the axis. The 

 binary axes mentioned are thus arranged like the infinitely low 

 steps of a spiral-staircase, be it dextro- or laevogyratory. There 

 are no planes of symmetry, nor a symmetry-centre present. If a cy- 

 lindrical rod be twisted by two equal but oppositely directed couples 

 at each of its ends, the whole system can be reckoned to have this 

 symmetry D^ . 



The group C has a heteropolar axis of isotropy A ^ , and an 

 infinitely great number of symmetry -planes passing through it. 



It has neither binary axes, nor a symmetry-centre. 



A truncated circular cone may be mentioned as an object having 

 this symmetry. Every vector which represents a force, a velocity, 

 etc., possesses the same symmetry; and it can be attributed also to 

 the electric current, or to the homogeneous electrostatic field of force. 



Finally the group C^ has no other symmetry -elements than a single 

 heteropolar axis of isotropy A ^ . 



An upright circular cone which is rotated round its axis with 

 a constant angular velocity in a definite sense, is an instance of 

 an object having this symmetry. In fig. 94. some schematical figures 

 will elucidate what is said here in the above. 



8. Although the five groups mentioned now possess, properly 

 speaking, an infinitely large number of non-equivalent symmetry- 

 properties, it can be easily understood, however, that the groups 

 Cg , C , and D^ only possess half, and C^ even no more than 

 a quarter of the symmetrical operations which are characteristic 

 of Dg, . They are related therefore to the last mentioned groups 

 as "sub-groups" are with respect to their "principal group", just 

 in the same way as hemihedral and tetartohedral crystal-classes are 

 related to their holohedral class of the same crystal-system. Indeed, 

 if by analogy, Z)g be considered as the holohedral class of the 

 "isotropous" system, Cg will represent the "pyramidal", D^, 

 the "trapezohedral" and C the "hemimorphic" hemihedrism 



