59 



are the same as several of the type C n . For if n is odd, C? is 

 evidently the same as C w ; and therefore in this case the symmetry 

 of the figure can be expressed by the symbol C n as well, as by Cjf. 

 Reviewing the above results, we may say: 



a. There are figures possible, whose symmetry is characterised 

 by the presence of a single axis A n of the first order, and by n planes 

 of symmetry passing through it. 1 ) The symbol of these groups is 

 C v tl \ their principal axis is a heteropolar one, as well as in the case of 

 the cyclic groups C n themselves. 



b. There are a number of figures, the symmetry of which consists 

 in the existence of a single homopolar axis A n of the first order, and 

 a plane of symmetry perpendicular to it. Their general symbol is C^f. // 

 n is odd, these groups are identical with C n for the same value of 



Fig. 60. 



Silver-iodide. 



Fig. 61. 

 Struvite. 



n; if n is even, they also possess a symmetry-centre, because they are 

 identical with the groups C J n for the same values of n. 



c. Other groups with one single axis A n of the first order are 

 impossible; for C is for n odd, identical with C^ n , and for n even, 

 with the groups C^. If, however, more axes of the second order were 

 present, the groups would possess more than a single axis, and 

 such groups of course do not belong to the kind here considered. 



8. It is of interest to look here for some representatives of 

 the discussed types of symmetrical objects, before we continue 



*) If one plane passes through an axis A n , there are n such planes passing 

 through it. This needs no further comment after what precedes. 



