Mr. W. Barlow on Crystal Symmetry. 



15 



Let A A' B B / (fig. 6) be lines drawn parallel to the two 

 axes through some point to meet a sphere described about 

 as centre, and let a. be the angle of the rotation-component of 

 the two identical screw-spiral movements about the two axes. 



Draw a great circle through A B K! B' and two other great 



circles through A A', B B' respectively to make angles ^ 



with A B A'' B' oppositely, the two latter circles intersecting 



in C, and let angle A C B be L Then, as shown by the 



familiar demonstration employed by Euler +, the change of 

 orientation effected by first carrying out the rotation a about 

 B B x and then the same rotation about AA', in the directions 

 indicated in the figure by the arrows, is that which is pro- 

 duced by a single rotation y about C in the direction also 

 there indicated. 



But, the change of orientation of a body which is effected 

 by a screw-spiral movement is that which would be brought 

 about by the component rotation acting alone without the 

 translation component. Therefore, since the rotations (a) 

 are the component rotations of the screw-spiral movements 

 about the two identical axes of the structure, the equivalent 

 rotation 7 about C is the rotation-component of the screw- 

 spiral movement which is the equivalent of the latter, and 

 which, as stated, is a coincidence-movement of the structure 

 in question. 



t See EntwicJcelung einer Theorie der Krystallstruktur, p. 31, iii. 



