24 Mr. W. Barlow on Crystal Symmetry, 



in this sphere the angular points of which are points of 

 intersection of these axes with the surface of the sphere. 



Proposition 20.— The existence of coincidence-axes of the 

 same rotation-angle intersecting the sphere of reference at the 

 angular points of a regular polyhedron* involves the existence 

 of axes centrally perpendicular to the polygonal faces and which 



have as one of their rotation- angles^ the angle </> = 2 . , 



ichere n is the number of sides of a polygonal face. 



For suppose, as before, that A t A 2 A 3 . . . A n are the angular 

 points of a polygonal face of n sides, and that 6 is the minimum 

 angle of rotation of the axes which pass through these points. 

 Then a coincidence-rotation 6 about A 2 which carries A 1 to A 3 

 carries the polygonal face A l A 2 A 3 . . . A n to the place of an ad- 

 joining similar face, and a similar rotation about A 3 (the new 

 position of Aj) which carries A 2 to A 4 leaves A 3 unmoved and 

 brings back the displaced polygon AjA 2 A 3 . . . A n to its original 

 position, but with A 1 moved to A 3 , A 2 to A 4 ; in other words, 

 the resultant of the two movements is a coincidence-rotation 



through an angle 2 . - about an axis through the centre of 



the polygon. 



When = 180°, and the angles of the polygon consequently 

 lie on a great circle, the rotation which brings A x to A 3 

 simply inverts the polygon so that, as a whole, it again 

 covers the same space ; the second rotation, that about the 

 axis through A 3 , annuls the inversion, leaving its plane as 



before, but rotated through the angle 2 . about an axis 



perpendicular to the polygon at its centre. 



Proposition 21. — 1 he only regular polyhedra whose angular 

 points depict the arrangement of a set of axes of one of the 

 possible kinds % in the way just explained are the cube, the 

 octahedron, and such of the limiting cases above referred to as 

 have for their two coincident faces regular polygons of 4, 6, 8, 

 or 12 sides. 



For the other regular polyhedra, which are, as is well known, 

 the tetrahedron, the regular-pentagonal dodecahedron, the 

 icosahedron, and such of the limiting cases as have not just 

 been enumerated, are inadmissible for the following reasons. 



(a) The tetrahedron because each of its four axes cuts the 



* As stated above, polyhedra of zero contents, having two coincident 

 polygonal faces lying in a great circle, are included under this definition. 



f This is not necessarily a minimum rotation-angle ; but if it is not, 

 it must be an integral multiple of a minimum rotation-angle. 



% See p. 17. 



