a2 Prof. Everett on Normal Piling as 
the cube-edges. This representation has the advantage that 
each cube, considered by itself, shows the positions of the 
12 spheres which touch a single sphere placed at the centre. 
Another representation is obtained by dividing space into 
parallelepipeds whose edges are equal and parallel to three 
edges of the fundainental. tetrahedron. The corners of the 
parallelepipeds will be the positions of the sphere-centres. 
Coming now to the subject of closeness of packing :-—It is 
possible to pack spheres in a box in a symmetrical arrange- 
ment in which each sphere touches 6 instead of 12, the points 
of contact being the ends of three diameters at right angles. 
The space thus occupied by a given number of spheres will 
be \/2 times as great as that occupied in normal piling. It 
_is also possible to adopt a symmetrical arrangement in which 
each sphere touches 8, the points vf contact being given by 
drawing lines from the centre of the sphere to the corners of 
the circumscribed cube. This will give a density of packing 
intermediate between the other two. 
It would, however, be a mistake to suppose that normal 
piling is the only system which gives the maximum of 
_ compactness. Every system of piling that has the maximum 
compactness must consist of parallel tiers in triangular 
arrangement ; and each sphere in a given tier must touch 
3 in each adjacent tier, besides 6 in its own; but these 
conditions allow, in adding each successive tier, a choice 
between two positions. In normal piling the choice is always 
made in the same way. In the system which comes next 
in order of simplicity, and which may conveniently be 
called antinormal piling, the choice is made alternately in 
one way and in the other. The departure from normal piling 
cannot begin till we come to the third tier: and if there are 
2+n tiers the number of different arrangements is 2”, all 
giving the maximum of compactness. 
In normal piling the choice is governed by a fixed tetra- 
hedron ; any two spheres that touch must be in a line parallel 
_to one of its edges. The alternative choice is represented by 
the edges of a second tetrahedron base-to-base with the first. 
The base is parallel to the tiers, and must not be turned about; 
but as regards the 3 edges which meet in the vertex, the 
change is the same as if they were rotated through 60° in the 
plane of the tiers. Hach sphere in a tier is surrounded by 6, 
giving 6 gaps 60° apart. Three spheres touching each other 
can be laid over 3 of the gaps, 120° apart ; and the question 
is, which of the 2 sets of 3 thus available shall we choose. 
In normal piling there are 6 directions of lines of touching 
spheres, 3 of them being in the tier planes, and parallel to 
