36 
from which we may determine the density. In the state 
Ai each sphere is in contact with 6 spheres, and the element 
of volume = Sr\ 
In the state A2 each sphere is in contact with 8 spheres, 
and the element of volume is 4^/3^^ 
In the state A3 each sphere is in contact with 12 spheres, 
and the element of volume is 4 \/ 
Secondly, shift the rows of each layer till the spheres in 
that layer occupy the position of maximum density, and 
then heat alternate layers as before, as denoted in the 
diagram B, and call the three states Bi, B2, B3. 
The state Bi will be identical with that called A2. In 
the state B2 each sphere will be in contact with 10 spheres, 
and the element of volume will be 6rl In the state Bg each 
sphere will be in contact with 12 spheres, and the element 
of volume will be 4^/2'^^, so that the states Ag and Bg have 
the same density, and each sphere has the same number of 
contacts. But the arrangements in each case will be 
different, and Bg admits of variety in the position of 
alternate layers. In order to show this, it is most con- 
venient to form what Listing calls the diagram of the 
complexus (“ Der Census raumlicher complexe.” Gottingen, 
1862), consisting in this case simply of the centres of the 
spheres. 
