25 



to that ot the ternary axis ..I tin- -econd order dealt with 

 in the above. 



Foi n c>, an analogous reasoning (fix. /./) shows, that thi- 

 e(|iii\-alent to a tcnhirv axis of the first order, combined witli an 

 <;/. Indeed, the arrow will successively reach the positions /, .?, 

 id -r, and 2, 4, and 6, so that e.g. 4 could also be obtained from /, 

 from 2, 6 from 3, etc., by simple inversion with respect to a centre 

 ietry 0. For n= 7 the result would have been analogous to 

 ise of n = 3, or n = 5; for = 8 however, we should have 

 )und an arrangement of the arrows, such as is represented in fig. is, 

 id here a-ain it appears that the complete symmetry of the set of 

 rows obtained, cannot be described by another combination of sym- 

 letry-elements, just as is the case when n is equal to 4. Later on 

 shall consider these cases in a more general way. For the present 

 will be sufficient to formulate the results obtained as follows here : 



Ln axis of the second order with a period of , is equivalent to an 



tis of the first order of the same period, combined with a r-eal 

 eflecting plane perpendicular to it, if n is an odd number. 



If however n is an even number, two cases must be considered: 

 r) if ;/ be divisible by 4: in this case the axis of the second order 



in never be replaced by another combination of symmetry- 



4* 



lents. And 2) when n is not divisible by 4 ( thus being odd): 



27T 



this case the axis of the second order with a period of is equi- 



M 



lent to an axis of the first order with a period-number = , combined 



ath a centre of symmetry. : ) 

 10. These two symmetry-elements, the axis of the first and 



lat of the second order, now having been considered in detail, 

 is of importance to notice here the result of repeated reflections 

 different mirror-planes, simultaneously present. It is supposed 

 this and all following cases, that the reflecting planes do not act 



From this it appears, that the centre of symmetry and the plane of 

 reflection are not sufficient to deduce all possible symmetries of those 

 Croups which only have axes of the first order. As soon as an axis has a 

 period whose number n is divisible by 4, the addition of a centre or of a 

 plane of symmetry can not lead to an exhaustive treatment of all possible 

 kinds of symmetry. Indeed, on account of this, Bra vais omitted in his famous 

 deduction of the possible groups of symmetry, the corresponding group of sym- 

 metrical polyhedra. 



