the Spinel Group of Crystals, 



307 



We begin with the oxygen atoms, of which there are four. 

 In order to maintain the trigonal symmetries these must lie 

 at the corners of a regular tetrahedron, so oriented that the 

 lines drawn from the centre to the corners of the tetrahedron 

 are parallel to the four diagonals of the crystal cube. 



Such an arrangement does not interfere with the sym- 

 metries about the (100) and (110) planes, as may be seen in 

 the following way. 



Consider first the case of the diamond, the structure of 

 which is shown in fig. 2 (loc. cit. p. 100). The black circles 



Fiar. 2. 



and white circles both represent carbon atoms, but the blacks 

 and whites can be considered separately as each forming 

 f;ice-centred lattices. They can be derived from one another 

 by a shift along a cube diagonal equal to one quarter of the 

 length of that diagonal. The symmetry about the plane egca 

 is not an absolute mirror symmetry ; but if all that is on the 

 right of that plane is reflected in it and then shifted by an 

 amount and in a direction represented for example by a 

 move from D half way to o, the reflected right coincides 

 absolutely with the left. White circles slip into the places 

 of black circles, but since both represent carbon atoms no 

 difference is made. 



Suppose now that the oxygen tetrahedra take the place of 

 the carbon atoms and are arranged, as already described, so 

 that the four perpendiculars from the corners on the faces 

 are parallel to the four cube diagonals. There are two ways 

 of making this arrangement. We may describe them by 

 saying that the coordinates of the four corners referred 

 to rectangular axes are in one case (111) (1,-1,-1) 

 (-1,1,-1) (-1,-1,1), in the other (-1,-1,-1) 

 (-1,1,1) (1,-1,1) (1,1,-1). 



Two tetrahedra so arranged are the reflexions ot each other 

 in the planes (xy), (yz), or (zx). 



X 2 



