63 



ilves, or may bisect the 



i 

 angle - 



n 



icrefore have a homopolar principal axis A n and binary axes 

 qtuatrd in a plan, perpendicular to A n , and are either homopolar, 

 it of two different sets, or heteropolar and of the same set (p. 38). 

 \\V must now add reflections 5 or an inversion / to the groups D n ; 

 every case the whole system of axes of D n must coincide with 

 1 1 M ! f by the operations corresponding 

 to the symmetry-elements added, 

 the following cases must be 

 into account : the added plane 

 reflection may be either horizontal : 

 or vertical S v , and this last may 

 through the binary axes them- 



Fig. 74- 

 A mphibole. 



itween two successive binary axes, 

 [n the first case we shall call it S v , in de other case S D ', to symbolise 



ic "diagonal" situation of it. Altogether we have now to discuss 

 the addition to D n pi the operations: S H , S v , S D , and /. 



The operations S H and S v are together equivalent to a rotation 

 through 180 round a binary axis, already found among the axes of 

 the group D n . Therefore in every case the result of combining 

 D n either with a horizontal plane of reflection, or with a vertical 

 me passing through a binary axis, will always be the same. 



However, if we combine S H and S D , the result will be equivalent 

 to a rotation round a binary axis, bisecting the angle between two 

 successive binary axes of the group D n already present. And as 

 such rotations are not yet included among those of the group D n , 

 the groups D^ and D% will be always different from each other. 



The combination of S n and / is equivalent to a rotation through 



50 round an axis coinciding with the principal axis A n . This 

 rotation is present or absent among those of D n , according as n is 

 m even or an odd number. Therefore, if n is even, D^ and D^ will 

 identical groups; only for n =-- odd number, the combination with 

 a symmetry-centre would produce a new group D* n , which might 

 appear different from any till now deduced. However on closer exa- 

 mination it becomes obvious that it is identical with the groups D% 

 already mentioned for odd values of n, because the inversion and 

 every binary axis together will produce a plane of symmetry per- 

 pendicular to the last one. We can thus include all cases in the 

 combinations of D n with S H and S D , and it is no longer necessary 



