28 Mr. W. Barlow on Crystal Symmetry. 



3. As to cases where the polygon is a square : — 



Here n = 4; <f> = 2 . ^! = 180°. 

 4 



In the simplest form there are therefore three digonal axes 

 at right angles to one another and no other axis, the digonal 

 axes being therefore all diverse — the type shown in (PL II.) 

 fig. VIII. (Gadolin, fig. 38). Every two of the three axes in 

 question intersect the sphere of reference at the four corners 

 of a square inscribed in a great; circle. 



With reference to the existence of any system possessing 

 the axes referred to and other axes in addition : — 



As the only point on the sphere whose distance from the 

 angles of the square is as great as a the side of the square, is 

 the point of intersection of the third axis (<j>) there can be no 

 additional digonal axes besides ihe three. Consequently the 

 only additional rotations possible are such as will bring the 

 system of three digonal axes to coincidence with itself. 



The axis of any additional rotation cannot coincide with 

 either of these three axes. For the only additional rotation 

 about one of these axes which would bring them to coin- 

 cidence is the conversion of a digonal into a tetragonal rota- 

 tion, and the existence of the latter is precluded in the case 

 in question because it would involve the existence of addi- 

 tional digonal axes. For the resultant of such a tetragonal 

 rotation and one of the existing digonal rotations is a digonal 

 rotation about an axis inclined at 45° to the existing digonal 

 axes*. 



The only other additional rotations which would bring the 

 system of digonal axes to coincidence are, it is evident, those 

 about axes equidistant from two or three of these axes 

 respectively. Axes equidistant from two are inadmissible 

 because they would be additional digonal axes. There re- 

 mains the case of additional trigonal axes equidistant from 

 three digonal axes. This case, which is therefore the only 

 additional one compatible with the presence of two digonal 

 axes so situated as to intersect the four angles of a square 

 in whose plane they lie, has been already deduced by another 

 method f. ^qo 



4. Where the polygon is a hexagon: n = 6; <j> = 2 . —rr- 

 = 120°. b 



In the simplest case there are therefore three digonal axes 

 in the same plane subtending angles of 60°, and a trigonal 



* The proof of this is substantially that given on page 24 of the value 

 of <£. 



t See above, p. 26 and (PL I.) fig. VI. 



