17 



On the other hand unlimited systems of points distributed in 

 space, as round. -red for instance in the theories of molecular 

 Structures, etc., are examples of infinite systems. 



\V< will MippiKe such an infinite system to be under investiga- 

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

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

 uted. no point whatever of the system will appear to remain 

 in its original position, in consequence of the translation, 

 is included in every helicoidal motion 1 ); however the figure 

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

 by saying that to every point of an infinite system an infinite 

 her of homologous points of the system always correspond, 

 a finite system be subjected to a helicoidal motion, the rotation 

 which corresponds to one characteristic of the figure under 

 tigation, it will reach a position such that a single translation 

 ukl bring it back to its original place ; by the motion con- 

 .ered, the figure comes into a new place in space, making it coincide 

 the figure itself. In the infinite system an infinite number 

 homologous points correspond to every point; in the finite 

 ure only a limited number. In the infinite system the translations 

 ;ntioned have thus a real importance with respect to the special 

 character of the unlimited symmetrical arrangement; in the finite 

 lix'iiv, however, those translations evidently are of no interest, as 

 long as the particular symmetry of the limited figure (polyhedron 

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

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

 have no significance as characteristic motions for finite systems ; 

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



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

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

 nslation, and the helicoidal motion also, may from the first be 

 luded as characteristic motions. For the definition of the special 

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

 -nppose only one point of the figure, the geometrical centre 

 mentioned previously, --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 



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

 (2). 45. (1907), Def. 7 and its Coroll. 8. 



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