Mr. W. Barlow on Crystal Symmetry. 25 



circle of reference in two points, and consequently the edge 

 of the cube outlined by the eight points o£ intersection thus 

 obtained, and not the edge of the tetrahedron, is the minimum 

 chord a between two axes*, and forms an edge of the 

 polyhedron. 



(b) The pentagonal dodecahedron because it gives n=o } 



and therefore cf> = 2 . = 144°. 



T n 



For the angles 2<£ = 288° and 360°- 2</> = 72° would, if <f> 

 has this value, also be rotation-angles of the axes in question, 

 and the value of 72° for the minimum rotation-angle is not 

 compatible with either of the minimum rotation-angles of 

 proposition 14, which are alone possible for coincidence- 

 axes. 



(c) The icosahedron because it gives 6 = 72°, which, as just 

 stated, is an inadmissible value for a rotation-angle. 



(d) In the limiting cases — those in which there are but 

 two polygonal faces which coincide with one another — 

 the value n = o is inadmissible because each of the three 

 digonal axes through the angles of the equilateral triangle 

 forming the two faces cuts the sphere in two points, and a, 

 the minimum chord between the axes, is therefore the side of 

 a regular hexagon and not the side of a triangle, i. e. when 

 there are three axes inclined at 120° the polygon which they 

 locate is a hexagon, not a triangle. Of other values for n all, 

 except n = 4, 6, 8, or 12, give an inadmissible value for the 

 angle of rotation </> of the principal axis of the system, since 



</> = 2 . — ; an d 60°, 90°, 120°, and 180° are the only values 



possible for minimum angles of rotation (Proposition 14). 

 Proposition 22. — In a system of axes: — 



(a) JSot more than one hexagonal axis can be present. 



(b) If there is a plurality of tetragonal axes the points of 

 intersection with the sphere of a set of them will lie at the six 

 angles of an octahedron. 



(c) If there is a plurality of trigonal axes, the angles of a 

 cube will mark the positions of a set of them. 



As to digonal axes, a much greater variety is possible. 



That there is no plurality of hexagonal axes is proved by 

 the fact that the axes of this order are not found among those 

 which locate the angles of the possible polyhedra in the way 

 explained in proposition 19. 



Tetragonal axes locate the angles of but one type of poly- 

 hedron — the octahedron; and therefore where more tetragonal 



* See above, p. 23. 



