68 



In this way we see the combination of an axis of the second order 

 appear, besides the planes of symmetry, within the scope of our 

 deductions; the combination mentioned evidently proves to have 

 significance only for an even period of the axis of the second order. 

 10. If, therefore, we review the results obtained by these consi- 

 derations, we can maintain generally, that all possible groups of the 

 second order which are directly related to the dihedron-groups of 

 the previous chapter, can be deduced from them by combination 

 with S H or S D , - - the last mentioned combination making the 

 principal axis A n simultaneously into an axis A zn of the second 

 order, with a period-number 2n. 

 Therefore : 



There are symmetrical figures which possess the axial system of the 

 groups D n , with a horizontal plane of symmetry perpendicular to the 

 principal axis A n , and thus containing all binary axes; moreover, they 

 possess n vertical planes of symmetry passing through A n and every 

 binary axis. If n is an even number, there will be also a symmetry-centre 

 present] if n is odd, however, the figure will have no centre of symmetry. 

 The symbol of these groups shall be D^. 



b. There are symmetrical figures which posses the axial system of 

 the groups D n , with a system of n vertical planes 

 of symmetry passing through 

 the principal axis A n , and 

 bisecting the angles between 

 every two successive binary 

 axes. If n is an even number, 

 the figure will have no sym- 

 metry-centre ; if, however, n is 

 odd, the group will also cer- 

 tainly possess such a centre. 

 In every case the principal 

 axis A n will be simultaneously 



Fg. 77. 



OH vine. 



an axis 



A 2n of the second order 



Fig. 78. 



with a period-number 2n. The symbol of these groups shall be 



including an angle of course, will be equivalent to a rotation round an 

 axis of the same direction as the principal axis A n of the group D n , but through 

 the double angle . The operation A 2 .S D of the new group is thus evidently 



equivalent to SH.A(), i. e. to the rotation round a mirror-axis with a period- 

 n 



number 2n. Thus the above-mentioned theorem is generally proved. 



