41 







Fig. 40. 



Iii these cases too it is obvious that all objects and figu 



this particular kind of symmetry, may take a second form 

 which is the mirror-image of the other. In the case of our stinvr, 

 ne would correspond to a right-handed, the other to a left- 

 mi K-d screw. 

 6. Up till now we have considered those figures which have 



ie axis of the period , or such j; 



ft 



possess two or more binary 

 ces. The only case yet remaining 

 therefore that, where the figure 

 more than one axis with a 

 jriod-number higher than 2. If 

 lis case too is treated in the most 

 ;neral way, we can really be sure 

 lat no other types of symmetry- 

 roups only having rotations round 

 ces of the first order, are omitted, 



id that, therefore, the question of the possible groups of this 

 :ind has been finally and exhaustively settled. 

 Let us suppose that a figure possesses rotations round an axis A 



the period , and also such round an axis B of the period . Re- 



ft ft 



lembering our previous conclusion that by the characteristic motions 

 the figure, it itself as well as its whole system of axes must be 

 ide to coincide with itself, it follows necessarily from this that 

 >und A there must be a number of n axes B equivalent to each 

 ther, and in the same way round B a number of p axes A , all of the 

 ie kind too. If a sphere with radius r be constructed round the 

 ted geometrical centre of the figure, the points ofi ntersection of 

 these axes B will be situated in the corners of a regular polygon 

 ,ith n sides, and those of the axes A in the corners of a regular 

 )lygon with p sides. As the whole system of axes must include a 

 lite number of them, it is evident that all these points must be 

 listributed over the whole surface of the sphere in such a way that 

 all these polygons are arranged as the faces of a regular polyhedron, 

 inscribed in the sphere, the regular polyhedron formed by 



all the points A being the polar figure of the regular polyhedron 

 formed by all points B as corners, and reversely. Now it is well- 

 known, that there are only five possible regular, endospherical poly- 



