Mr. W. Barlow on Crystal Symmetry. 23 



of the coincidence-axes the points of intersection of some set of 

 similar axes in the sphere of reference with the spherical sur- 

 face will mark the angular points of one or other of the regular 

 polyhedra inscribable in the sphere ; including, however, in this 

 category the extreme cases of polyhedra of zero contents ivhose 

 faces are two coincident regular polygons inscribed in a great 

 circle. 



For suppose A l5 A 2 (fig. 8) are the positions on the sphere of 

 some two axes of the same rotation-angle 6 so selected as to be at 

 a minimum distance apart measured 

 by a chord a. Then the carrying out of 

 the rotation about A 2 will discover 

 a third axis A 3 , identical with A l7 and 

 chord A 1 A 2 = A 2 A 3 =ffl, and the sphe- 

 rical angle A 1 A 2 A 3 = 6 I . Similarly 

 other axes A 4 A 5 . . . A n of the same 

 rotation-angle can be discovered all 

 lvinor on the circle drawn through 

 Aj A 2 A 3 , the alternate axes being 

 moreover identical. 



One or other of the axes thus located 

 will coincide with A 1# For suppose the 

 process be continued till we reach an 

 axis A«, whose distance from A l5 measured along a chord as 

 before, is not > a; then, since a is the minimum distance apart 

 of two axes of the same sort, the distance A n A x must =a and 

 A n+1 be coincident with A x . 



Thus the set of axes Aj A 2 A 3 . . . A n map out the angles of a 

 regular polygon of n sides on the surface of the sphere *. 



A second polygon identical with the first, but which will 

 be distinct from it except when = 180°, is located by a 

 similar series of rotations in the opposite direction. The 

 angles of this second polygon will also lie on the surface of the 

 sphere. 



Again: — As the rotation-angle 6 is an aliquot part of 360° 



(Prop. 11), let 0= ; there are thenjt? identical polygons 



of the nature just traced grouped around the axis A x without 

 overlapping ; and as the same will be true with regard to 

 each of the axes which has been located, each polygon will be 

 completely surrounded by polygons identical with it which do 

 not overlap it or one another. Therefore in all cases, where 

 more axes than one of the same rotation -angle are present in 

 the sphere of reference, a regular polyhedron can be inscribed 



* Comp. ftohneke, Entwickelung einer Theorie der Krystallstruktur, 

 Satz. xi. p. 42. 



