. Nature . ther than the one thus derived. The systemis therefore termed the cubic closest-packed assemblage ofequal spheres, and, being derived in the manner described,still retains the high symmetry of the cube ; the fragmentshown, in fact, outlines a cube. Three directions at rightangles in it, those which are parallel to the three cubeedges, are seen to be identical in kind ; this identity in kindin the three rectangular directions a, 6, and c is con-veniently expressed by the ratio a:b : c = i : i : i. On removing spheres from one corner of the cubicclosest-packed assemblage of equal sphe
. Nature . ther than the one thus derived. The systemis therefore termed the cubic closest-packed assemblage ofequal spheres, and, being derived in the manner described,still retains the high symmetry of the cube ; the fragmentshown, in fact, outlines a cube. Three directions at rightangles in it, those which are parallel to the three cubeedges, are seen to be identical in kind ; this identity in kindin the three rectangular directions a, 6, and c is con-veniently expressed by the ratio a:b : c = i : i : i. On removing spheres from one corner of the cubicclosest-packed assemblage of equal spheres a closetriangularly arranged layer is disclosed, and, by similarlytreating each corner of the fragment of assemblage, thecube outline gives place to one of octahedral form. Theassemblage is now seen to be built up by the superpositionof the disclosed triangularly arranged layers, the hollowsin one layer serving to accommodate the projecting partsof the spheres in adiacfiit layers. When this operation is. performed it is perceived, however, that two ways of stack-ing the layers homogeneously are possible. The first ofthese, in which the fourth layer lies immediately over thefirst, the fifth over the second, and so on, yields the cubicclosest-packed assemblage. The alternative mode of stack-ing, in which the third layer lies immediately over the first,the fourth over the second, and so on, exhibits the samecloseness of packing as the first, but possesses the symmetryof the hexagonal crystal system ; it is accordingly termedthe hexagonal closest-packed a,ssemblage of equal spheres(•ig- 4)- of the hexagonal assemblage showsthat the horizontal directions, in the planes of the layers,are not identical in kind with vertical directions perpen-dicular to the planes of the layers. Correspondingdimensions in these two directions, a and c, are in theratio of a:c = i: ^/(i)= i:0-8165. The final step in the treatment of the closest-packedassemblages of equal spheres consis
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