90 



A x 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. 92 some schematical figures 

 will elucidate what is said here in the above. 



8. Although the five groups mentioned now possess, properly 

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

 properties, it can be easily understood however that the groups 

 CQO > ^oo ' an d A only possess half, and C^ even no more than 

 a quarter of the symmetrical operations which are characteristic 

 of D^ . 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, D* be considered as the holohedral class of the 

 "isotropous" system, C^ will represent the "pyramidal", D ( 

 the "trapezohedral" and C the "hemimorphic" hemihedrism 

 of that system, while C^ may be considered to be a "tetartohedral" 

 class of it. P. Curie pointed already to this analogy of the groups 

 considered with those of the ordinary crystal-systems. x ) 



9. Now we must draw closer attention to the question: how 



) P. Curie, Bull, de la Soc. Miner. 7. 418, a. f . (1884). 



