38 



axes: one corresponding to OL 2 , the other to OL' Z , the last 

 axes being the bisectors of the angle between every two successive 

 axes of the first set, etc. The truth of this can easily be seen from 

 figure 2p, where m 4 : by turning it round the perpendicular to 

 the plane of the drawing N, it may be clear that only the 



axes a-iNa'i and a 2 Na' 2 will 

 coincide, and in the same 

 way b^Nb'i and b 2 Nb' 2 . Here, 

 moreover, it appears that 

 both en,ds of every axis will 

 "CL Z coincide with both ends of the 

 other axis of the same set: 

 thus e. g. Na with Na 2 , 

 Na\ with Na' 2 , etc. 



If however the number m 

 characteristic for the axis 

 Fig. 30. perpendicular to the plane of 



the drawing, is not an even, 



but an odd number, only one end of each axis will coincide with 

 one end of each of the other ones: thus in fig. jo, where m = j, Na l 

 with Na 2 and Na 3 , but Na\ only with Na' 2 and Na' 3 respectively. 

 This is often expressed by saying that in the last case the binary 

 axes are heteropolar, although they all belong to the same set, in 

 contrast with the case first mentioned. There they were homopolar, 

 however, the binary axes belong at the same time to two dif- 

 ferent sets. 



The principal axis ON must of course be always homopolar, because 

 binary axes perpendicular to it are present. 



If we review the results obtained up till now in the cases considered, 

 we can conclude therefore: 



There are groups of symmetry, characterised by a principal ho- 

 mopolar axis ON, with a period , n being 2 or greater than 



rv 



2, and 'by n binary axes situated in a plane perpendicular to ON 



7T 



and intersecting at angles of . These binary axes are homopolar, but 



belong alternately to two different sets if n is an even number', and 

 the axes arc equivalent but heteropolar if n is an odd number. The 

 corresponding groups are named dihedron-groups, and they will in future 

 generally be denoted by the symbol D n . 



