282 Mr. L. Fletcher's Crystallographw Notes. 



Fig. 3 represents a second crystal, with its faces a x b x e x d x 

 u ifiiyi§i parallel respectively to the faces abcdot/3y$ of 

 fig. 2. 



Both versions of the law assume that the plane of rotation 

 or the twin-plane is a face of the form {101}. Let us take 

 for the particular plane of rotation the plane (101) which 

 truncates the edge 8 X e lf or the edge 8 c, and is represented in 

 the stereographic projection by its pole T. 



On rotating the crystal represented in fig. 3 through two 

 right angles round the normal T T to the plane (1 1), its 

 faces will take up positions represented in fig. 4; and the poles 

 of the faces in this new arrangement are introduced in fig. 1 

 as o x b\ Cj d x a x j3 x 7i 8 X . In the first place we may remark that 

 as each of the pairs of faces 8 X c x and y x d x is diagonally sym- 

 metrical to the line T T about which the rotation takes place, 

 C\ will after the rotation have a direction parallel to that belong- 

 ing previously to the face 8 X , and still belonging to the face 8 ; 

 and similarly the face d x will after the rotation have a direction 

 parallel to that belonging previously to the face y x and still 

 belonging to 7. In other words, the faces c x d x y x 8 X of the 

 rotated octahedron shown in fig. 4 are respectively parallel to 

 the faces 8ydc of the octahedron of fig. 2. If the octahe- 

 dron had been the regular one of geometry, not only these 

 faces, but all the remaining faces and all the edges of the octa- 

 hedron of fig. 4 would have been parallel to faces and edges 

 of the octahedron shown in fig. 2. As the line T T bisecting 

 is perpendicular to the edge C A, and therefore not 

 parallel to the edge CA, the point T will not be midway 

 between A and C ; and thus, although the edge A C would 

 be unchanged in direction by a rotation through two right 

 angles about T T, while B would be rotated to B and B to B, 

 yet C would not be rotated exactly to A nor A exactly to C ; 

 and the edge C A of fig. 2 will thus not be parallel to A x C x 

 of fig. 4, norjhe edge B A to the edge B x Cj. In fact, while 

 the edge A x C, is parallel to the edge C A, the angle A x C x A x 

 is, as stated above,_90° 51', and the angle C AC is 89° 9'; 

 whence the edges C, A x C A must have a mutual inclination 

 of 1° 42'. Similarly, although the edge A, U x is parallel to 

 the edge C A, and the plane C A B to the plane A x O x B x , the 

 angle A x C x B x , as stated above, is 60° 2_9/,_and the angle 

 C A B is 59° 45I 7 , whence the edges B A B x <J X are mutually 

 inclined at an angle of 43^. 



The same results will follow, perhaps more simplv, from 



