CHAPTER II. 



General Considerations on the Change of Position of stereome- 

 trical Figures. Characteristic Motions. - - Figures and their 

 Mirror-images. Reflection and Inversion. Finite and Infinite 

 Figures. Symmetry-Properties, Symmetry-Elements of the First 

 and Second Order. Euler's Theorem. Deduction of Symmetry- 

 Character as a Mathematical Problem. - - Geometrical Centre of 

 Finite Figures. Periods of Axes of the First and of the Second 

 Order. - Special Cases. - - Repeated Reflections. - - Demon- 

 stration of the general Symmetry -Relations. 



1. If a stereometrical figure F be brought from its original 

 position in space 5 X into a different position S 2 , two cases will be 

 distinguished. The first case is, that the transition from 5 X to 5 2 , 

 can be made by means of a motion, i. e. by a translation (a shift 

 parallel to itself), by rotation, or by helicoidal motion, this being a 

 combination of the two former. In the positions S l and S 2 , the 

 figure thus remains congruent with itself. This could also be regarded 

 as if two congruent figures F were compared, but in two different 

 positions S 1 and 5 2 . As a corollary, therefore, it must always be 

 possible to bring two congruent figures F into coincidence by mere 

 motions, if they have initially different positions in space. 



Now we will suppose that the figure jp is a symmetrical one, in 

 the sense of our definition in the previous chapter. Then it will always 

 be possible to make such a choice of the motions mentioned, that 

 the figure can be brought from its successive positions S 2 , S 3 , 5 4 , etc., 

 to self-coincidence or in its original place in space by mere trans- 

 lations. If this is the case, we will call the motions performed as 

 characteristic of the particular symmetry of the figure F. 



An example will make this clear. 



Let A (fig. 8) be a cube, the corners of which for the sake of 



