20 



as these, as possessing only symmetry -properties of the first order: 

 the characteristic motions are rotations through definite angles a, 

 and round definite axes, and will be known as axes of symmetry 

 of the first order. These axes of symmetry are therefore named the 

 special symmetry -elements of the first order. 



If, however, the figure is of such a nature, that it is equivalent 

 to its mirror-image in several ways, and if here also the point 

 is supposed to remain fixed in space, we have already seen that 

 besides the symmetry-properties of the first order, there must also 

 be introduced other symmetry-properties by which the figure is 

 changed into its mirror-image. It is then said to possess also sym- 

 metry-properties of the second order \ and as has already been demon- 

 strated, the characteristic operation corresponding to these, will 

 generally consist in definite rotations around a certain axis, insepa- 

 rably combined with a reflection in a plane perpendicular to that 

 axis. 1 ) This species of axis will be discriminated by us in the 

 following pages as an axis of symmetry of the second order, or a 

 mirror -axis. The mirror-axis is the characteristic symmetry-element 

 of the second order, just in the same way as the ordinary symmetry- 

 axis is for symmetrical figures of the first order. 



If in fig. 12 A 4 is a mirror-axis with the characteristic angles of 

 rotation 90, 180, 270, the arrow will get the positions indicated, 

 if subjected to the four characteristic operations essential for the 

 axis ]4 4 . It is obvious that the inversion and the ordinary reflection 

 in a mirror-plane are only special cases of the mirror-axis : for a = 

 we have the pure reflection, and for a = 180, as was demonstrated 

 above, we have the inversion. 



7. In a wellknown theorem of elementary mechanics, which is 

 also named after Euler, it is proved that, if two rotations 

 around two intersecting axes are executed successively, they are 

 together equivalent to a rotation round a third axis, passing through 

 the point of intersection 2 ). 



From this it follows that, if a symmetrical figure possesses two 

 characteristic rotations round the axes of symmetry A and B inter- 



1 ) The rotation around the axis and the reflection in a plane perpendicular to 

 it have no significance here independently of each other: only the result of their 

 combined action must be taken into account. 



2 ) A simple demonstration is given at the end of this chapter, as a corol- 

 lary of a general theorem by Boldyrew. 



