22 



Now it can be demonstrated *), and the method will be briefly 

 shown later on, - - that the operation resulting from the presence 

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

 a translation differing from zero. In the same way it can be shown 

 that, if a figure has two axes of symmetry of the second order, which 

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

 From these few examples it will be seen, that such cases cannot 

 occur in finite symmetrical figures, the helicoidal motion there 

 being excluded for reasons already given. 



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

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

 order should pass through the same fixed point 0. In the same way the 

 planes of reflection must pass through it, and if an inversion-centre 

 be present, that must also coincide with this point 0. It will be dis- 

 criminated, as previously said, as the geometrical centre of the figure 

 F; it may play the role of an inversion-centre (centre of symmetry) 

 also, but this need not always be 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, 

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

 reflecting planes, etc. 



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

 sider the finite symmetrical systems apart from the infinite ones. 



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

 detail the general character of the axes of symmetry of the first 

 and second order and to examine more in particular the results 

 of repeated reflections in several planes, before the possible combina- 

 tions of symmetry-elements are systematically discussed. For these 

 investigations appear to be of great importance for the purpose 

 of 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 

 is known when its characteristic angle of rotation & is given. That 

 angle is defined as the smallest angle through which the symmetrical 



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 29 

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



