§ 26. If we look from above at the solids of which Fig. 12 shows the outline, we see essentially a hollow leading down to a perforation between _S_{ 1}_, _S_{ 2}_, _S_{ 3}_, and if we look from below we see a hollow leading upwards to the same perforation: this for brevity we shall call the perforation _pqr_. The diagram shows around _S_{ 1}_ six hollows leading down to perforations, of which two are similar to _pqr_, and the other three, of which _p′q′r′_ indicates one, are similar one to another but are dissimilar to _pqr_. If we bring _S_{ 4}_ from above into position to touch _S_{ 1}_, _S_{ 2}_, and _S_{ 3}_, its place thus found is in the hollow _pqr_, and the places of all the solids in the layer above that of the diagram are necessarily in the hollows similar to _pqr_. In this case the solids in the layer below that of the diagram must lie in the hollows below the perforations dissimilar to _pqr_, in order to make a single homogeneous assemblage. In the other case, _S_{ 4}_ brought up from below finds its place on the under side of the hollow _pqr_, and all solids of the lower layer find similar places: while solids in the layer above that of the diagram find their places in the hollows similar to _p´q´r´_. In the first case there are no bodies of the upper layer in the hollows above the perforations _similar_ to _p´q´r´_, and no bodies of the lower layer in the hollows below the perforations _similar_ to _pqr_. In the second case there are no bodies of the upper layer in the hollows above the perforations _similar_ to _pqr_, and none of the under layer in the hollows below the perforations _similar_ to _p´q´r´_.
§ 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_.