Diamond with Theoretical Carbon Atoms. 261 



If we at first consider only those atoms in planes -f 2, zero, 

 and —2, it is always true that those atoms nearest to form 

 a regular geometrical figure with fourteen faces, six squares, 

 and eight equilateral triangles, one carbon a torn being located 

 at each of the points 1-12 inclusive. These atoms are in 

 pairs, each pair being on opposite sides of at equal dis- 

 tances on some diameter through 0. An inspection shows 

 that the axes of rotation of each atom in one pair are parallel, 

 as 1 and 4. The axis of 1 points away from the centre J of 

 the tetrahedron 0123, and of 4 toward the centre of 0456, 

 and these directions are parallel. The axes of the atoms in 

 the hexagon 7-12 inclusive point in pairs toward the centres 

 of the tetrahedra whose bases lie outside the hexagon, 7 and 



8 toward 13, which is the centre of a tetrahedron on base 7, 

 8, 16 ; 9 and 10 toward 14, the centre of a tetrahedron on 

 base 9, 10, 17 ; 11 and 12 toward 15, the centre of a tetra- 

 hedron on base 11, 12, 18. Therefore 7 and 10, 8 and 11, 



9 and 12 are parallel to each other in pairs. 



It is evident that the turning moment of the triangle of 

 atoms 1, 2, 3, on is zero, for the moment of 1 on may be 

 represented by a vector perpendicular to the plane 1J0, which 

 contains both the axes of 1 and 0. Similarly, the moment of 

 2 on is represented by a vector perpendicular to plane 2J0 

 and equal in amount, and the moment of 3 on by a vector 

 perpendicular to plane 3 JO of equal amount. These three 

 vectors take the directions 0, 8 ; 0, 10 ; and 0, 12 respec- 

 tively, making 120° with each other in the same plane, and 

 the sum is, therefore, zero. 



It is seen from this that the sum of the moments of all the 

 atoms from 1 to 12 inclusive is zero, for these twelve atoms 

 form three squares 1, 4, 8, 11 ; 2, 5, 7, 10; and 3, 6, 9, 12; 

 the axes in the first square all being parallel to the axis of 1, 

 in the second square parallel to 2, and in the third parallel 

 to 3. The total moment of the twelve atoms is, therefore, 

 four times the moment of: the triangle 1, 2, 3, namely, zero. 

 It may also be shown that these moments produce stability 

 for small displacements of the axis of 0, but these are not all 

 the atoms which must be considered. The triangle of atoms 

 1, 2, 3 lies in the plane we have called No. + 2, the centre 

 of the triangle being a point where the tails of the three 

 arrows meet just to the left of in fig. 2. A study of this 

 figure shows that the total moment of all the atoms in this 

 plane, + 2, is zero. By drawing concentric circles in this 

 figure from a centre where the tails meet it is easy to show 

 that the sum of the moments of all atoms in each circle is zero. 

 The next outlying atoms to 1, 2, 3, in this plane form an 



