38 



are simultaneously present, and begin with the simplest case of this 

 kind, i. e. when two binary axes L 2 and L 2 intersect in the geome- 

 trical centre O of the figure (fig. jo) at an angle 0. In fig. 30 the 

 axes L 2 and L 2 are situated in the plane of the drawing ; ON 

 may be the perpendicular to this plane in O. Because L 2 is a 

 binary axis (# = 180), a rotation round L 2 will simply interchange 



both ends of the line ON, its 

 lower and upper parts being 

 reversed by it. If now a rota- 

 tion round L 2 occurs through 

 180, both ends of ON will 

 interchange once more, ON 

 therefore returning finally to 

 its initial position. Thus ON 

 must be itself an axis of rota- 

 tion, characteristic as a sym- 

 metry-element of the figure 

 considered. Its typical angle 

 of rotation a can be easily 

 found. For if OZ be a straight line of the figure situated in the 

 plane (L 2 OL 2 ), this line will reach the position OZ' by the rotation 

 round OL 2 , and OZ" by the rotation round OL 2 ; it has thus changed 

 from OZ to OZ", the angle ZOZ" being equal to 20. Since 

 the same transition would n 



v%/ 5 



occur if the figure were rotated 



round the axis of symmetry . \)[ b 



ON through its characteristic 



angle a, this angle must be a 



equal to 20 also. We can 



therefore conclude from this : a| 



// a finite symmetrical figure 

 possesses two binary axes inclu- 

 ding an angle 0, it possesses 

 also an axis of symmetry with 



\b, 



the characteristic angle 20, 

 perpendicular to the plane of 

 the binary axes. Fig 31 



However, we can go yet 



farther. For it must be evident that if a figure F has a system 

 of symme try-axes, every characteristic rotation round one of these 



