Mr. W. Barlow on Crystal Symmetry. 27 



But if one digonal axis be made tetragonal, the existence 

 of the remaining axes involves the necessity of all the three 

 digonal axes being so converted; and if one digonal axis 

 be drawn to bisect opposite cube-edges, the presence of five 

 other digonal axes bisecting the remaining ten edges of the 

 cube is similarly established. 



Further, one of the tetragonal and one of the trigonal 

 rotations, when combined, have for resultant one of the added 

 digonal rotations which bisect cube-edges*. Consequently 

 the introduction of the tetragonal rotations involves that of 

 the six digonal rotations — they are not independent of each 

 other. 



Therefore, finally, where the polyhedron traced is a cube 

 there is but one type which has additional coincidence- 

 movements besides those of the type last given, viz. the type 

 given in (PL II.) fig. VII. (Gadolin, fig. 27), in which the three 

 digonal axes have become tetragonal and new digonal axes 

 have appeared which bisect all the cube-edges. In this case 

 too, for the reason stated in the last example, the axes of 

 each set are all identical. 



2. As to the octahedral arrangement: — 



"Where the polyhedron traced by a set of axes of the same 



rotation-angle is an octahedron: n = o; = 90°; <£ = 2.— ^— 



= 240°. And a rotation-angle of 240° involves one of 

 360 o -240 o =120°. The axes necessarily involved by the 

 existence of this form of polyhedral arrangement are there- 

 fore three tetragonal axes which intersect opposite vertices 

 of the octahedron and four trigonal axes which pass through 

 the centres of opposite faces. 



This type is identical with that last traced ; digonal axes 

 bisecting the 12 octahedron-edges are identical with those 

 bisecting the cube-edges in the last example, and are, as 

 stated, involved by the presence of the other axes. A similar 

 argument to that above employed shows that no other axes 

 are possible where a polyhedron traced by the axes in the 

 way described is an octahedron. 



Next come the limiting cases of polyhedra of zero contents 

 with two coincident polygons as faces. In all these 0=180°, 

 i. e. the axes which contain the angular points of the polv- 

 hedron are always digonal ; they lie in the plane of the 

 polygonal faces. 



For the proof of a very similar proposition see p. 24. C'omp. 

 Soimcke's Entivickehmg einer Theorie der Krystillstruktur, p. 31. 



