39 



axes must not only bring the figure F into coincidence with itself, 

 but also the whole system of axes. If this were not the case, the 

 group of symmetry-properties could not be a finite group. If now 

 we make the characteristic rotations round ON, it is clear that we 

 shall find in the plane (L 2 OL 2 ) several more binary axes, making 



r\ 



with each other angles of 2$, where 20 = , m being the integer 



indicating the period of the axis ON. 



In the same way we shall see that there are two sets of such binary 

 axes: one corresponding to OL 2 , the other to OL' 2 , 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 3 1 , where m = 4 : by 

 turning it round the perpen- 

 dicular to the plane of the 

 drawing N, it will be clear 

 that only the axes a l Na{ 

 and a 2 Na 2 will coincide, and 

 in the same way b Nb{ and 

 b 2 Nb 2 . Here, moreover, it 

 appears that both ends of 

 every axis may coincide with 

 both ends of the other axis 

 of the same set: thus e. g. 

 Na l with Na 2 , Na{ with Fig. 32. 



Na 2 , etc. 



If, however, the number m characteristic for the axis 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. 32, where m = j, Na l with Na 2 and Na 3 , 

 but Na( only with Na 2 and Na' 3 successively. This is often expressed 

 by saying that in the last case the binary axes are heteropolar, 

 although they all belong to one set, in contrast with the case first 

 mentioned. There they were homopolar, the binary axes belonging at 

 the same time to two different sets. 



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

 binary axes perpendicular to it are present. 



Reviewing the results obtained so far in the cases considered, 

 we may conclude as follows: 



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



