30 Mr. W. Barlow on Crystal Symmetry. 



Therefore the two types described are the only ones com- 

 patible with the polygonal arrangement o£ digonal axes in 

 question. 



5. Where the polygon traced is an octagon : n = 8; 



^=2.^=00°. 



o 



In the simplest case there are therefore four digonal axes 

 lying in the same plane and a tetragonal axis at right angles 

 to these; the arrangement being shown in (PL II.) fig. X. 

 (Gadolin, fig. 32). 



To prove tbat no other case of this polygonal arrangement 

 of digonal axes is possible, i. e. that no additional axis or 

 additional rotation about an existing axis can be present : — 



First, it may be observed that the addition of a digonal 

 axis in the plane of the four digonal axes of the simplest case 

 is precluded because the chords a, which are the sides of the 

 octagon, are minima. 



Next, that the addition of any other axis than this would 

 involve a plurality of tetragonal axes. Now if more than one 

 tetragonal axis is present, there must be three altogether 

 which are placed at right angles to one another, as shown by 

 proposition 22 *. Two of these would therefore lie in the 

 plane of the four digonal axes. But a tetragonal axis placed 

 anywhere in this plane is sufficiently near to some digonal 

 axis to involve the presence of four digonal axes whose 

 distance apart is under the minimum a. Consequently there 

 can be no tetragonal axis or any other axis in any position 

 not previously occupied. 



As the essential tetragonal axis does not admit of con- 

 version because an axis whose order is a multiple of it is 

 inadmissible, no axis whatever can be added. 



b*. Where the polygon traced has twelve sides: n = 12; 



<£=2.^ = 60°. 



In the simplest case there are therefore six digonal axes 

 lying in the same plane and a hexagonal axis at right angles 

 to these; the arrangement being shown in (PL II.) fig. .XL 

 (Gadolin, fig. 44). 



Since no additional digonal axis in the plane of the six 

 essential digonal axes is admissible, and the presence of any 

 other additional axis anywhere else would involve the exist- 

 ence of a plurality of hexagonal axes, which according to 

 proposition 22 is impossible, there can be no other case of this 

 polygonal arrangement. 



* See p. 25. 



