26 



For n = 6, an analogous reasoning (fig. 16) shows that this axis 

 is equivalent to a ternary axis of the first order, combined with an 

 inversion. Indeed, the arrow will successively reach the positions i, j, 

 and 5, and 2, 4., and 6, so that e.g. 4. could also be obtained from i, 

 5 from 2, 5 from j, etc., by simple inversion with respect to a centre 

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

 the case of n = j, or n = 5; for n = 8 however, we should have 

 found an arrangement of the arrows, such as is represented in fig. 17, 

 and here again it appears that the complete symmetry of the set of 

 arrows obtained cannot be described by another combination of sym- 

 metry-elements, just as is the case when n is equal to ^. Later on 

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

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



O 



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



axis of the first order of the same period, combined with a real 

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



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

 i) if n be divisible by 4: in this case the axis of the second order 

 can never be replaced by another combination of symmetry- 

 elements; and 2) when n is not divisible by 4, I thus being odd): 



2w 

 in this case the axis of the second order with a period of is 



fl 



/yt 



equivalent to an axis of the first order with a period-number - 

 (period: 1, combined with a centre of symmetry. l ) 



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

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

 it is of importance to notice here the result of repeated reflections 



x ) From this it appears that the centre of symmetry and the plane of reflection 

 are not sufficient to deduce all possible symmetries of those groups 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, Bravais in his famous deduction of the possible groups of symmetry 

 ("Etudes cristallographiques" , 2ieme Partie, pag. 129) introduced the "plane of 

 alternating symmetry", which has the same function as our axis of the second 

 order. However, in his opinion, the corresponding group of symmetrical polyhedra 

 had no practical significance for crystalline substances, its occurrence in nature 

 being most improbable, if possible at all. (Cf. loco cit., pag. 130, 176). 



