Wl 



21 



ill happfii. it tin IM- simultaneous axes of symmetry do not int< i 

 in>taii< < it they cross in space? 



it ran In <1< monstrated 1 ), and the method will be briefly 

 own lahT on, -- that the operation resulting from the presence 

 two crossing axes of rotation must be a helicoidal motion, with 

 nslation differing from zero. In the same way it can be shown, 

 t it ,1 figure has two axes of symmetry of the second order, which 

 not intersect, the resulting motion will also be helicoidal; etc. 

 m these few examples it will be seen that such cases cannot 

 ur in finite symmetrical figures, the helicoidal motion there 

 ing excluded for reasons already given. 



n the case of finite symmetrical figures therefore, it is strictly neces- 

 that all possible symmetry-elements of the first and of the second 

 should pass through the same fixed point 0; thus also the planes 

 reflection must eventually pass through it, and if an inversion 

 tre be present, that must also coincide with this point 0. It 

 be discriminated, as previously said, as the geometrical centre 

 t tin figure F; eventually it may play the role of an inversion-centre 

 (crntre of symmetry) also, but this need not be always the case. 

 However, infinite figures may certainly have symmetry-elements, 

 not passing through one and the same point simultaneously. In 

 such unlimited systems there may be present parallel, intersecting, 

 d crossing axes of the first or of the second order, sets of parallel 

 ecting planes, etc. 



In respect to the foregoing therefore it seems necessary also to con- 

 .er the finite symmetrical systems apart from the infinite ones. 

 9. In this connection it seems advisable to consider in some 

 ail the general character of the axes of symmetry of the first 

 d second order and to examine more in particular the results 

 repeated reflections in several planes, before the possible combina- 

 >ns of symmetry-elements are systematically discussed. For these 

 estigations appear to be of great importance for the purpose 

 understanding the doctrine of symmetry in general, and for the 

 demonstration of its theorems. 

 a. Axes of symmetry of the first order. 



The axis of symmetry of the first order is in each case determined 

 by its special direction in space, and by its own character which 



1) All these theorems are gone into thoroughly by A. K. Boldyrew, Verh. 

 der Kais. russ. Miner. Ges. St. Petersburg, (2). 45. (1907); vid. theorems 2 

 till 38, and 25 till 28, and also the problems 7 till 11 in his paper. 





