66 



In this connection it is of importance to draw attention to a 

 special property of the axis A n if such a dihedron-group D n is made 

 into one of the second order by adding a diagonal mirror-plane 

 S D to it. It can easily be proved by means of group-theoretical ar- 

 gumentations that in this case the axis A n is transformed at the 



*] 



same time into an axis A zn of the second order l ) with a period of . 



rt 



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. 



9. 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 n 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 be 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 



*) For the general and simple demonstration of this theorem, the same 

 symbols for the "multiplication" of operations of the first and second order 

 can be used as we drew attention to previously. Let SD be the diagonal 



plane bisecting the angle !L between two successive binary axes of D n , and let 



n 

 AZ be a rotation through 180 round such an axis ; S// and SV may be positions 



of planes of reflection, as we have defined them in 8 of this chapter. 

 Then we have: A 2 = SH.SV, and therefore A 2 .SD = SH.SV.SD. Now sr.SV, 

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



271 



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 .SD of the new group is thus evidently 

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

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



