58 



As we have seen, we can use for that purpose either the reflection 

 in a plane, or the inversion, because the simultaneous presence of 

 several axes of the second order always involves the coexistence 

 of rotations, and thus the groups of this kind can be reduced to 

 the cases in which these rotations are combined with reflections or 

 with the inversion. For if not so, the simultaneous addition of several 

 axes of the second order to a rotation-group, would in general imply 

 the formation of other axial combinations than those already deduced 

 in the preceding chapter, and this is impossible. The axes of the 

 second order in groups of the second order, if present therein at all, 

 can, therefore, only coincide with the axes of the first order, because 

 each axis of the second order is partially also one of the first at the 

 same time. The only question is therefore: in what way must these 

 planes of reflection or this symmetry-centre be combined with C n ? 



Of course this must happen in such a way, that the whole axial 

 system of the group will coincide with itself by the operation 

 which results from the addition of the new symmetry-element. In 

 the case where only a single axis A n is present, as in our groups C n , 

 this can evidently be the case only if the added plane of symmetry 5 

 be either perpendicular to the axis A n , or passes through that axis. 



If we suppose A n to be in a vertical position, we can indicate both 

 kinds of reflections by S H (horizontal reflecting plane) and by S v 

 (vertical reflecting plane), and we have now only to investigate, if the 

 groups of the second order thus obtained : C^,C^,and in the case of the 

 addition of the symmetry-centre : C^, are identical or different groups. 



To answer the last question we have simply to investigate what 

 will be the result of the combination of the operations S H , S v , and 

 7, taken two at a time. Now S H and S v together will be equivalent 

 to a rotation through 180 round a horizontal axis; also S v and 

 / combined. But the combination of S 7/ and 7 will be equivalent 

 to a rotation through 180 round a vertical axis, and this operation 

 will be present or not present among the rotations of C n , ac- 

 cording as n itself is either an even or an-odd number. If, therefore, 

 n is an even number, the combination of C n with S H or with 7 

 will give identical results : in this special case the groups C 1 ^- 

 are identical with C^, according to the theorem mentioned above 

 (p. 53). If, however, n be odd, we shall have three kinds of new 

 groups of the second order. 



But in connection with what was said in the discussion of the 

 groups C n , it will be obvious that some of the groups here considered 



