Suppose now a homogeneous assemblage of the given bodies, in open order with no contacts, to be arbitrarily made according to any nine arbitrarily chosen values for the six distances between a point of _S_{1}_ and the corresponding points of its six pairs of nearest and next nearest neighbours (§ 1 above), and the three angles (§ 9 above) specifying the orientation of each body relatively to rows of the assemblage. We may choose in any nine rows through _S_{1}_ any nine pairs of bodies at equal distances on the two sides of _S_{1}_ far or near, for the eighteen bodies which are to be in contact with _S_{1}_. Hence there is an infinite number of solutions of the problem of which only a finite number can be real. Every solution of the problem of eighteen contacts is imaginary when the shape is wholly convex. [Illustration: FIG. 13.] § 36. Without for a moment imagining the molecules of matter to be hard solids of convex shape, we may derive valuable lessons in the tactics of real crystals by studying the assemblage described in §§ 24 and 25 and represented in Figs. 12 and 13. I must for the present forego the very attractive subject of the tactics presented by faces not parallel to one or other of the four faces of the primitive tetrahedrons which we found in § 24, and ask you only to think of the two sides of a plate of crystal parallel to any one of them, that is to say, an assemblage of such layers as those represented geometrically in Fig. 12 and shown in stereoscopic view in Fig. 13. If, as is the case with the solids[10] photographed in Fig. 13, the under side of each solid is nearly plane but slightly convex, and the top is somewhat sharply curved, we have the kind of difference between the upper and under of the two parallel sides of the crystal which I have already described to you in § 21 above. In this case the assemblage is formed by letting the solids fall down from above and settle in the hollows to which they come most readily, or which give them the stablest position. It would, we may suppose, be the hollows _p′ q′ r′_, not _p q r_, (Fig. 12) that would be chosen; and thus, of the two formations described in § 25, we should have that in which the hollows above _p′ q′ r′_ are occupied by the comparatively flat under sides of the molecules of the layer above, and the hollows below the apertures _p q r_ by the comparatively sharp tops of the molecules of the layers below.