18 



that the whole figure remains in its original place in space during 

 all motions and reflections which are characteristic of its symmetry. 



It may here be emphasized once more, that among the number of 

 their characteristic motions, infinite symmetrical figures always 

 possess translations too, and such figures can therefore eventually 

 have helicoidal motions also. For several of such systems, especially 

 for those which play an important role in the theories of crystalline 

 structure, such helicoidal motions are really typical. 



6. For the time being we can leave the discrimination between 

 the two cases of finite and infinite figures, and proceed with our 

 task of characterising the various typical operations for the deter- 

 mination of their possible symmetries. 



An arbitrary stereometrical figure, of which one point remains 

 fixed in space, can always be brought from a position S l into another 

 position 5 2 , where-in it -is congruent with itself, by a single and 

 completely determined rotation round an axis A, passing through 

 the fixed point 0. This is the well-known theorem of Euler 2 ), 

 by which all rotatory motion in elementary mechanics can be treated 

 in a very simple way. 



It follows from this, that the most general characteristic motion 

 of a finite symmetrical figure which is congruent with itself in several 



!) The theorem of Euler can easily be proved, as soon as the validity 

 of the thesis is accepted that two rotations round two axes A and B inter- 

 secting in 0, are together always equivalent to a third rotation round an axis C, 

 passing throught O also. The demonstration of this is given later on. Now, 

 if the validity of this theorem be accepted, we can demonstrate the theorem 

 of Euler easily. For let the figure F now be brought from its original position 

 Si into a final position 82, a point of it O remaining fixed in space. One of 

 the straight lines of F, e.g. OLj may be brought into its new position OL% by the 

 said transition. We imagine a plane passing through OLi and OLg, and consider 

 the normal N there-on in ; the directions OLi an d OLg may include an angle a ; 

 If now the figure F be rotated round N over an angle a, OLi comes into 

 OZ-2, and the new position of F is Si". To bring it from Sj" to Sg, we have 

 only to rotate it round OLg ; for OLg has in S# the same position as it has 

 now, its points thus remaining fixed in space, and those therefore being points 

 situated on a true ,,axis" of rotation. 



The whole transition from S l to S 2 therefore can be considered to be 

 equivalent to the rotations round N and OL 2 , and these are equivalent to a 

 rotation round some axis C. The problem to find this third axis C, if the 

 positions of two others are given, will be gone into at the end of this chap- 

 ter, after the general method of reasoning by means of repeated reflections 

 has been described. 



