56 



Perhaps some representatives of the family of the radiolaries, 

 as e. g., Aulosphaera elegantissima, may be conjectured to possess 

 this symmetry. 



6. Proceeding with the deduction of the possible groups of 

 the second order, we can. now start with those groups C n of the 

 first order dealt with in the previous chapter, which only possess 

 a single heteropolar axis of the first order, and combine these 

 groups C n with a typical symmetry-element of the second order 

 in the way formerly discussed. 



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 can be 

 reduced to the cases in which these rota- 

 tions 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 al- 

 ready deduced in the preceding chapter, 

 and this is impossible. The axes of the 



Fig 56. 



Copper-sulphate 



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

 all, can therefore only coincide with the axes of the first order, be- 

 cause each axis of the second order is at the same time always 

 also one of the first order. 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 

 Hhe 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 S 

 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 5 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. 



