24 



with them, because A z is an axis of the second order. Then we shall 

 find the arrow repeated six times in such positions in space, that 

 the complete set of the six arrows thus obtained has a symmetry 

 which can also be described by the presence of a ternary axis of the 

 first order and a real reflecting plane perpendicular to it. This can 

 easily be seen from a figure or a suitable model. 



lin = 4 however, we shall find in the same way, that the complete 



Fig. 14. 



A 



Fig. 15. 



set of different positions reached by the arrow is that represented 

 in the fig. 13. Although A^ appears to be also an axis A^ of the 



27T 



first order with a period , it is evidently not possible now 



to substitute A by other symmetry-elements which can comple- 

 tely describe the particular symmetry of the figure thus obtained. 

 iLFor n = 5 we shall find, on examination, that the axis of the second 

 order A 5 is also an axis A 5 of the first order, combined with a plane 

 of symmetry perpendicular .to it. This case is thus evidently wholly 



