66 



latter case may pass through the binary axes themselves, or may 

 bisect the angle between two successive binary axes. In the first 



case we shall call it S v , in the other case S D , to symbolise the 

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

 addition to D n of the operations: S H , S F , S D and 7. 



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 

 one passing through a binary axis, will always be identical. 



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 7>f will be always different from each other. 



The combination of S H and 7 is equivalent to a rotation through 

 180 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 

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

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

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

 appear different from any hitherto 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 

 any 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 

 to consider the combination with 7. Although we might stop here, 

 as the combinations with S H , S D , S v , and 7 have now been 

 sufficiently discussed, it may yet be of interest to extend these 

 discussions. Of course it will then appear, that really no new groups 

 can be produced beyond those already mentioned. 



For this purpose let us first investigate the combination of S r 

 and S D . This combination will be equivalent to a rotation round 

 an axis A n through an angle which is double that between S v and S D , 



i.e. through an angle . As this rotation is not yet included among 

 those characteristic of D n , -- because the angle of rotation corres- 

 ponding to A n is , the groups D% and D% will really be diffe- 



