24 



with respect to their particular period. However, it must be dis- 

 tinctly remarked, how a closer examination will soon prove that in 

 many (not in all) cases, axes of the second order can be replaced 

 by those of the first order, if combined with a real reflecting plane or 

 with a centre of inversion. It is therefore our task to investigate, 

 when this is possible and when not. Two cases of this kind have 

 already been dealt with : the binary axis of the second order (n = 2) 



Fig. 14. 



Fig. 15. 



was equivalent to the inversion, and in the case, where n = 1, the 

 axis was equivalent to the reflection in a real plane. The first 

 value of n to be examined is thus n = 3. Let ~A Z be a ternary axis 

 of the second order (fig. 14) and let us consider, which positions, 

 the arrow / wills uccessively reach. Let us execute all rotations round 

 Zg and combine them with the reflections inseparably connected 

 with them because ^A 3 is an axis of the second order. T hen we shall 

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



