37 



Fig. 28. 



n axis of rotation, characteri-tn .1- a symm-trv-rlc-m<-Mt oi tin- 

 considered. Its typical angle of rotation a, can be easily found. 

 or if OZ be a straight line of the figure situated in the plane (L 2 , 

 this line will reach the position OZ' by the rotation round 

 'L a , and finally OZ" by the rotation round OL 2 '; it has thus come 

 cm OZ to OZ", the angle 

 ()/" being equal to 2$. 

 ce the same transition 

 uld occur if the figure 

 re rotated round the axis 

 f symmetry ON through its 

 aracteristic angle , this 

 ngle must be equal to 2$ 

 so. We can therefore con- 

 lude from this: 

 // a finite symmetrical figure 

 ossesses two binary axes 

 ing an angle <p, it 

 possesses also an axis of symmetry with the characteristic angle 2<p, 

 erpendicular to the plane of the binary axes. 

 However, we can go yet farther. For it must be evident that if 

 figure F has a system of symmetry-axes, every characteristic 



rotation round one of these 

 axes must not only bring the 

 figure F into coincidence with 

 itself, but also the whole 

 system of axes. If this were 

 not the case, the group of 

 symmetry-properties could 

 OL, not be a finite group. If now 

 we make the characteristic 

 rotations round ON, it is 

 clear that we shall find in 

 the plane (L 2 OL'^ several 

 more binary axes, making 

 with each other angles of 2$, 



where 2$ = , m being the 



a 



i 



Fig. 29. 



integer indicating the period of the axis ON. 



In the same way we shall see that there are two sets of such binary 



