50 



If we remember its true characteristic operation, and the fact that 

 two successive reflections in the same plane always neutralise each 

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

 equivalent operations: A(&), A(2a), A(ja).... etc., only 14 (<*), 

 A(3a), A(jtx). . . . etc., can be true operations of the second order, 

 transforming the symmetrical figure F into its mirror-image F' '. 



The others: Z(<?#), A~(4&), ~A(6u) etc , will be simple rotations 



round the axis A n , through angles 20,, 4%, etc.; that their number is 

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

 number, needs no further comment. 



From this consideration it is at once evident that there must 

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

 an even number. For if n is an odd number, the operation A(not) will 

 be equivalent to a mere reflection of the original figure in a plane 

 perpendicular to A n , because not = 360, and the figure is trans- 

 formed by this odd number of operations of the second order into its 

 mirror-image. The operation A [(n + 1)^], therefore, will be equivalent 

 to a pure rotation of the initial figure, and thus will not be 

 comparable with any of the operations among the n first terms 

 of the series. It follows from this, that if n is an odd number, 

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

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

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

 ~A(nofy\ and in consequence of a general theorem which will be 

 discussed in 3, the symmetry of the whole group may be defined 

 also as the combination of certain rotations with reflections in a 

 plane perpendicular to the axis A n . 



If, however, n be an even number, we may at once conclude from the 

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

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

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

 It is also obvious that there is now no pure reflection present 

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



/yj 



here, may be an odd or an even number. 



If n is even, but odd, evidently the term of the series A I -a J 

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

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

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



