49 



\\v remember iN true < harai tei i-tic operation, and the fact that 



successive reflections in the same plane always neutralise each 



it must be immediately evident, that in the series of non- 

 jiiivalent opei-ations: A(), A(2x), A(jtx>).... etc., only A(), 

 i A(s*).... etc., can be true operations of the second order, 

 >rming the symmetrical figure F into its mirror-image /'. 

 le others: A(2x), A (40), A(6<x).... etc., will be simple rotations 

 >und the axis A n , through angles 2<x, 40, etc.; that their number is 

 idrntly tin same as that of the former ones, if n be an even 

 iber, needs no further comment. 



1 rom this consideration it is at once evident that there must 

 a difference between the cases when n is an odd., and when an 



en number. For if n is an odd number, the operation A(na) will 

 . mere reflection in a plane perpendicular to A n , because not = 

 ', and the figure is transformed by this odd number of operations 

 the second order, into its mirror-image. The operation A [(n + \)a\ 

 icrefore will represent a pure rotation, equivalent to A(ot), and thus 

 ill not be equivalent to any of the operations among the n first 

 ;rms of the series. It follows from this that if n be an odd number, 

 icre are in the whole series 2n non-equivalent operations present, 

 which n are mere rotations, and n true operations of the second 

 ler. Among these last ones there is present a pure reflection (namely 

 {(not); and in consequence of a general theorem which will be 

 :ussed in 3, the symmetry of the whole group may be defined 

 as the combination of certain rotations with reflections in a 

 me perpendicular to the axis An- 

 il however n be an even number we may at once assume from the 

 resence of the axis A n , that there are only n non-equivalent ope- 

 itions in the complete series. Half of them are rotations, and the 

 ther half are true, non-equivalent operations of the second order, 

 is also obvious that there is now no pure reflection present 

 long the last ones; but two different cases must be distinguished 



re, may be an odd or an even number. 



If n be even, but - odd, evidently the term of the series A\- 

 will be a pure inversion, because it corresponds to the combination 

 of the rotation through an angle -a =: 180, with a reflection in a 



plane perpendicular to it; and as we have seen in the theorem on 



4 





