BarLtow—A Mechanical Cause of Homogeneity of Crystals. 569 
which, while they may approximate to a general uniformity 
of distribution, cannot reach homogeneity, and are unable to 
arrive at stable equilibrium. 
Now, although it would appear that these two divisions 
embrace all possible cases of assemblages whose dimensions are 
large, a little consideration shows us that for thin assemblages, #.e., 
those in which either one or two of their three dimensions are 
small, slightly impaired homogeneity, not incompatible with stable 
equilibrium within the assemblage, may give closest-packing. 
The following will make this clear in a single instance :— 
If in a cubic partitioning of space spheres of one size be placed 
at the cube centres, and spheres of another size, bearing a certain 
proportion to the first, at the cube angles so as to form a stack 
such as is described on p. 549 (fig. 9), this stack may be regarded 
as made up of triangularly-arranged layers, each composed of 
one kind of sphere only; the planes of these layers are perpendi- 
cular to some one of the four directions of the cube diagonals, and 
alternate layers are composed of spheres of the same size. 
Suppose now that instead of spheres of two sizes only, four 
different sizes are used, and, instead of an infinite number of layers 
but four consecutive triangularly-arranged layers are present. It 
is then evident that if we preserve about the same proportion 
between the sizes of succeeding layers as prevails in the stack just 
referred to, but make the spheres of the third layer avery little smaller 
than those of the first, and those of the fourth with a similar relation 
to those of the second, closest-packing will be reached, not when the 
centres of the equal spheres of a layer lie in the same plane, but when 
they lie on some curved surface very approximately spherical, the four 
different surfaces traced by the centres in the different layers being 
practically concentric. Since the curvature of the layers is very 
slight, as compared with the sizes of the spheres, the departure from 
uniformity in the arrangement of a layer caused by its following 
a curved surface instead of a plane will be but trifling, even in the 
ease of layers of considerable extent. 
Further, if, instead of one set composed of four layers, the 
assemblage consists of several such sets, it is evident that an effect 
of the same kind may be looked for, although in this case, for 
the arrangement to be congruent, the conditions in each curved 
