51 



page 16, this combination is always equivalent to an inversion. As a 

 consequence of the same general theorem in 3 mentioned above, 

 we shall see that the symmetry of the whole group may, therefore, 

 be also described in this case as the combination of an axis of the 



4?r 

 first order A n , with a period of , and a symmetry-centre. 



Y n 



The special example of n 6 in Chapter // was an illustra- 

 tion of this. 



But if - be itself an even number, (i. e. if n itself is divisible by 



2 __ / ^ . 



four), the term A f -<*) is now equivalent to a simple rotation round 



the axis A n through an angle of 180. Now, as there is present among 

 the operations of the group neither a pure reflection nor an 

 inversion, the axis of the second order A n can in this case not be 

 replaced by any combination of other symmetry-elements. 



Some simple instances may make these facts more evident. If 

 n = 4, we have the four non-equivalent operations: A(po), A(i8o), 

 ^(270), and A(j6o). From these the second and the fourth are equi- 

 valent to pure rotations through 1 80 and 360 respectively, because 

 the figure is brought into positions, in which it is congruent with 

 itself. But A(po) and ~A(2?o) are true rotations round a mirror- 

 axis. None of those four operations, however, is either a pure reflection 

 or an inversion, and therefore the symmetry of the figure cannot 

 be described in this case as a combination of an axis A z or A 

 with some symmetry-element of the second order. If n = j, there 

 are six non-equivalent operations: A(i2o), A(2^.o) t A(j6o), 

 A(4&o), A(6oo), and ^(720). From these the first, third, and 

 fifth operations are true operations of the second order; but A(24O) 

 ~A(4.8o), and ~A(?2o) are mere rotations round an axis A 3 through 

 angles of 240, 120, and 360 respectively. Among the three opera- 

 tions first mentioned, A(j6o) is evidently equal to a pure reflection 

 in a plane perpendicular to A 3 . The whole symmetry of this group 

 can therefore, according to the theorem dealt with in 3 below, be 

 described as the combination of a ternary axis A 3 and a plane of sym- 

 metry perpendicular to it. Evidently in the complete discussion 

 of each case not only the angle of rotation must be taken into 

 account, but also the special place which the operation considered 

 occupies among the complete set of successive operations of the 

 series: those with even number of succession are always pure 

 rotations. 



