Mr. W. Barlow on Crystal Symmetry. 



11 



intersect some space-unit, has corresponding to it some line 

 drawn in the other corresponding direction which similarly 

 intersects some other space- unit ; therefore some one of 

 those screw-spiral movements of the group which are applic- 

 able to the selected unit will carry the one line to the initial 

 place of the other line. 



Now the translation component of this movement does not 

 affect direction, and therefore it is only the rotation com- 

 ponent which is concerned in bringing the direction of the 

 one line to that of the other. 



But any screw-spiral movements which have the same 

 rotation component, i. e. whose axes are parallel and rotate 

 through the same angle in the same direction, produce 

 identical changes of orientation. Consequently it is only 

 those with different rotations that, from a given unit, locate 

 differently orientated, units and the corresponding differently- 

 directed lines which intersect these units. All directions 

 which have identically the same relation to the structure as 

 some given direction, will therefore be located by successively 

 applying to the latter the different coincidence rotations 

 applicable to some single space-unit. 



Proposition 4. — The symmetry of arrangement of the 

 various directions found identically related to the homogeneous 

 structure of a mass is that of identically placed lines drawn 

 through the centre of some body so formed as to possess axes 

 passing through its centre whose directions and rotations are 

 identical to those referred to in the last proposition, and no 

 others. Thus for every homogeneous structure in which two 

 or more directions are identical, a sphere of reference can be 

 constructed with axes through its centre, so that the repetition 

 of identical directions on this sphere corresponds exactly to 

 that prevailing in the homogeneous structure. 



Proof. By hypothesis the axes of the centred body have 

 the same directions and rotations as those of the structure. 

 Therefore every direction in the structure has corresponding 

 to it a line in the centred body which makes the same series 

 of angles with the axes of the latter as the direction in 

 question makes with the axial directions of the structure. 

 Apply to a selected direction, i. e. to some line drawn in this 

 direction, the various movements of the structure so as to 

 locate other lines or directions identically related to the 

 structure, and apply to the corresponding line in the centred 

 figure the rotations of this figure. 



As the changes of orientation produced by the latter 

 rotations are the same changes of orientation as those brought 

 about by the movements of the structure and no others, thy 



