32 



as it is, has some disadvantages in so far, that in such figures as 

 differ from their mirror-images, such reflections can only have a 

 virtual significance, these figures of course possessing no real plane 

 of symmetry. 



This fact may cause some confusion, especially to students to 

 whom these reasonings are new. But as a mere mathematical method, 

 the conception mentioned may be of general use; and it is of 

 importance to recognise this fact when special theorems relating 

 to the general symmetry of stereometrical figures have to be 

 strictly demonstrated. 



In the next chapters we shall now proceed to the final deduction 

 of all the possible combinations of symmetry-elements, and to a 

 rational classification of them for the purpose of morphological 

 description in general. 



here. Therefore in the case where an axis of the first order is replaced by 

 two intersecting mirror-planes including an angle a, only half the number of 

 points produced by the successive reflections must be taken into account; and 

 where the axis of the second order is replaced by the cooperation of three 

 mirrors, the third of which is perpendicular to both the others, only a fourth 

 of the points produced by the reflections must be considered in these deduc- 

 tions. Wulff therefore distinguishes the action of such combined mirrors as 

 hemi-, resp. tetarto-symmetry. We shall call the mirror-planes real planes of 

 symmetry, if all points produced in the successive reflections are taken into 

 account; in all other cases the reflecting planes have only virtual significance for the 

 symmetry of the figure considered. 



