Mr. W. Barlow on Crystal Symmetry. 21 



rotation than a ; and this is inconsistent with a being the 

 minimum rotation. 



Thus in the simplest possible class of cases — those in which 

 the axes present in a structure have but one single direction, 

 the nature of the directional symmetry of the structure is not 

 affected by the relative arrangement of these axes, and will be 

 fully expressed by a sphere of reference which has a single axis 

 whose minimum rotation is the least rotation-component of 

 any of the coincidence- movements of the structure ; e. </., a 

 structure whose axes are hexagonal, trigonal, and digonal all 

 parallel to one another, will have its identically related direc- 

 tions arranged just as they are found in a sphere of reference 

 which has a single hexagonal axis *. 



These conclusions may be summed up in the following 

 proposition regarding the symmetrical arrangment of direc- 

 tions which are identically related to the structure where the 

 homogeneity is of the nature above defined : — 



-Proposition 17. — All the different types of symmetrical 

 arrangement of corresponding directions are comprised in the 

 series of types of centred symmetry which present every con- 

 ceivable kind of identical repetition about a centre consistent 

 with the presence of only such axes as are possible for the 

 structures in question ; i. e., no other axes than digonal, trigonal, 

 tetragonal, or hexagonal f. 



So far as identical repetition of parts and directions is con- 

 cerned, the task remaining to be done consists therefore first 

 in obtaining the complete series of types of centred symmetry 

 having the axes referred to, and then in making sure that some 

 homogeneous structure or other is conceivable corresponding 

 to each of them : in other words, in ascertaining that none of 

 these types is impossible for homogeneous structures. 



Cases where the Awes all have the same Direction. 



The least complicated types of repetition of identical 

 directions are those in which the coincidence axes present all 

 have the same direction, and, before passing to cases in which 

 there is a plurality of axial directions, the following simple 



* Similar reasoning to that given above, shows that if a structure 

 possesses axes of more than one sort parallel to one another it also possesses 

 axes with a rotation which is the difference between the rotations of 

 such axes ; e . g. a structure with digonal and trigonal axes parallel to one 

 another must also possess hexagonal axes. 



t See prop. 14, p. 17. 



