31 



I lii-> tact may cause some confusion, especially to students to 

 /horn these reasonings are new. 



I'.ut .1- .1 mere mathematical method, the conception mentioned 

 lay be of general use; and it is of importance to recognise this 

 K t . in rase some special theorems relating to the general symmetry 



stereometrical figures have to be strictly demonstrated. 



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



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

 itional classification of them for the purpose of morphological 

 Ksniption in general. 



each other, but only the "final effect" of their cooperation is considered 

 sre. Therefore in the case where an axis of the first order be replaced by 

 iro intersecting mirror-planes, including an angle x, only half the number of 

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

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

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

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

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

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

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

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

 ictry of the figure considered. 



