18 



space, as considered for instance in the theories of molecular 

 structures, etc., are examples of infinite systems. 



We will suppose such an infinite system to be under investiga- 

 tion, and let the most general type of motion, the helicoidal one, 

 be in some way characteristic of it. If this helicoidal motion is 

 executed, no point of the system will appear to remain in space 

 in its original position, in consequence of the translation which 

 is included in every helicoidal motion l ) ; however, the figure as, 

 a whole remains at the same place in space. This is also 

 expressed by saying that to every point of an infinite system 

 an infinite number of homologous points of the system always 

 correspond. If a finite system be subjected to a helicoidal motion, 

 the rotation of which corresponds to one characteristic of the 

 figure under investigation, it will reach a position such that a 

 single translation would bring it back to its original place; by 

 the motion considered, the figure comes into a new place in 

 space, making it parallel to itself. In the infinite system an 

 infinite number of homologous points correspond to every point; 

 in the finite figure only a limited number. In the infinite system 

 the translations mentioned have thus a real importance with respect 

 to the special character of the unlimited symmetrical arrangement; 

 in the finite figure those translations are evidently of no interest, 

 as long as the particular symmetry of the limited figure (polyhedron 

 e. g.) is regarded as being defined by its characteristic motions or 

 reflections. From this we can safely conclude, that helicoidal motions 

 can have no significance as characteristic motions for finite systems; 

 only those need be considered here, the translations of which are 

 equal to zero, i. e., when they are simple rotations about an axis. 



Thus for the description of the symmetry of finite figures, the 

 .translation, and the helicoidal motion also, may from the first be 

 excluded as characteristic motions. For the definition of the special 

 symmetry-character of such figures it thus appears to be sufficient 

 to suppose only one point of the figure, the geometrical centre 

 previously mentioned, --to remain fixed in space during all sym- 

 metrical operations to which the figure may be subjected. As 

 already said, it is always possible to choose this point so 

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



*) Vid: A. K. Boldyrew, Verb, der Kais. russ. Miner. Ges. St. Petersb., 

 (2). 45, (1907), Definition 7 and its Corollary 8. 



