19 



is only the rotation round an axis. We shall define such figures 

 as possessing only symmetry-properties of the first order: 

 r -liar.ii t listic motions are rotations through definite angles , 

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

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

 ial symmetry-elements of the first order. 



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

 iN mirror-image in several ways, and if here too the point 

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

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

 introduced other symmetry-properties by which the figure is 

 iiM'd into its mirror-image. It is then said to possess symmetry- 

 erties of the second order too; and as already has been demon - 

 ted, the characteristic operation corresponding to these, will 

 erally consist in definite rotations about a certain axis, insepa- 

 ly combined with a constantly repeated reflection in a plane 

 ndicular to that axis. *) This remarkable species of axis will 

 discriminated by us in the following pages as an axis of symmetry 

 the second order, or a mirror-axis. The mirror-axis is the charac- 

 istic symmetry-element of the second order, just in the same way 

 as the ordinary symmetry-axis is for symmetrical figures of the 



order. 



If in fig. 10 AI is a mirror-axis with the characteristic angles of 

 ation 90, 180, 270; the arrow will give the positions indicated, 

 subjected to the four characteristic operations essential for the 

 is A. It is obvious that the inversion and the ordinary reflection 

 a mirror-plane are only special cases of the mirror-axis : for a. = 0, 

 h.-ive the pure reflection; and for a 180, as was demonstrated 

 >ve, we have the inversion. 



7. In a well-known theorem of elementary mechanics, which is 

 named after Euler, it is proved that if two rotations 

 und two intersecting axes are executed successively, they are 

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

 e point of intersection. 2 ). 

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



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

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

 combined action must be taken into account. 



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

 lary of a general theorem by Boldyrew. 



