57 



an>wer the last (|iiestion we have -imply to investigate what 

 e the result ,,t th<- eombination o! the opei,:i and 



taken two at a time. Now S 11 and S y together will !) ( <iui\alrnt 

 a rotation through 180 round a horizontal axis; S y and / com- 

 ied, too. Hut the combination of S v and 7 will be equivalent 

 a rotation through 180 round a vertical axis, and this ojx-ration 

 lx- pn-sent or not present among the rotations of C n , ac- 

 mling as n is either an even, or an odd number itself. If therefore 

 be an even number, the combination of C n with S H or with I 

 ill give identical results: in this special case the groups C% 

 identical with C*. according to the theorem mentioned above, 

 however n be odd, we shall have three kinds of new groups of 

 ic second order. 



But it will be obvious, in connection with what was said in the 

 :ussion of the groups C n , that some of the groups here considered 

 iv the same as several of the type C n . For if n be odd, C% is 

 /idently the same as C n ; and therefore in this case the symmetry 

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

 To sum up the above results, we can say: 



a. There are figures possible whose symmetry is characterised 

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

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

 ; their principal axis is a heteropolar one, as well as in the case of 

 cyclic groups C n themselves. 



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

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



plane of symmetry perpendicular to it. Their general symbol is C". If 

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

 if n be even however, they also possess a symmetry-centre, because 

 are identical with the groups for the same values of n . 



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

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



fiih the groups C%. If more axes of the second order were however 

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



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

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



the discussed types of symmetrical objects, before we continue 



l ) Jf one plane passes through an axis A n , there are n such planes passing 

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





