§ 27. Going back now to operation (1) of § 23, remark that when the point of contact _p_ is arbitrarily chosen on one of the two bodies _S_{1}_, the point of contact on the other will be the point on it corresponding to the point or one of the points of _S_{1}_, where its tangent plane is parallel to the tangent plane at _p_. If _S_{1}_ is wholly convex it has only two points at which the tangent planes are parallel to a given plane, and therefore the operation (1) is determinate and unambiguous. But if there is any concavity there will be four or some greater even number of tangent planes parallel to any one of some planes, while there will be other planes to each of which only one pair of tangent planes is parallel. Hence, operation (1), though still determinate, will have a multiplicity of solutions, or only a single solution, according to the choice made of the position of _p_. Henceforth however, to avoid needless complications of ideas, we shall suppose our solids to be wholly convex; and of some such unsymmetrical shape as those indicated in Fig. 12 of § 25, and shown by stereoscopic photograph in Fig. 13 of § 36. With or without this convenient limitation, operation (1) has two freedoms, as _p_ may be chosen freely on the surface of _S_{1}_; and operation (2) has clearly just one freedom after operation (1) has been performed. Thus, for a solid of any given shape, we have three disposables, or, as commonly called in mathematics, three ‘independent variables,’ all free for making a homogeneous assemblage according to the rule of § 22. § 28. In the homogeneous assemblage defined in § 24, each solid, _S_{1}_, is touched at twelve points, being the three points of contact with _S_{2}_, _S_{3}_, _S_{4}_, and the three 3’s of points on _S_{1}_ corresponding to the points on _S_{2}_, _S_{3}_, _S_{4}_, at which these bodies are touched by the others of the quartet. This statement is somewhat difficult to follow, and we see more clearly the twelve points of contact by not confining our attention to the quartet _S_{1}_, _S_{2}_, _S_{3}_, _S_{4}_ (convenient as this is for some purposes), but completing the assemblage and considering six neighbours around _S_{1}_ in one plane layer of the solids as shown in Fig. 12, with their six points _prq″p′r′q″′_ of contact with _S_{1}_; and the three neighbours of the two adjacent parallel layers which touch it above and below. This cluster of thirteen, _S_{1}_ and twelve neighbours, is shown for the case of spherical bodies in the stereoscopic photograph of § 4 above. We might of course, if we pleased, have begun with the plane layer of which _S_{1}_, _S_{2}_, _S_{4}_ are members, or with that of which _S_{1}_, _S_{3}_, _S_{4}_ are members, or with the plane layer parallel to the fourth side _S_{2}_ _S_{3}_ _S_{4}_ of the tetrahedron: and thus we have four different ways of grouping the twelve points of contact on _S_{1}_ into one set of six and two sets of three.