Consider a solid _S_{1}_ and the twelve neighbours which touch it, and try if it is possible to cause it to touch more than twelve of the bodies. Attach ends of three thick flexible wires to any places on the surface of _S_{1}_; carry the wires through interstices of the assemblage, and attach their other ends at any three places of _A_, _B_, _C_, respectively, these being any three of the bodies outside the cluster of _S_{1}_ and its twelve neighbours. Cut the wires across at any chosen positions in them; and round off the cut ends, just leaving contact between the rounded ends, which we shall call _f′f_, _g′g_, _h′h_. Do homogeneously for every other solid of the assemblage what we have done for _S_{1}_. Now bend the wires slightly so as to separate the pairs of points of contact, taking care to keep them from touching any other bodies which they pass near on their courses between _S_{1}_ and _A_, _B_, _C_ respectively. After having done this, thoroughly rigidify all the wires thus altered. We may now, having three independent variables at our disposal, so change the orientation of the molecules, relatively to rows of the assemblage, as to bring _f′f_, _g′g_, and _h′h_ again into contact. We have thus six fresh points of _S_{1}_; of which three are _f′_, _g′_, _h′_; and the other three are on the three extensions of _S_{1}_ corresponding to the single extensions of _A_, _B_, _C_ respectively, which we have been making. Thus we have a _real_ solution of the interesting geometrical problem:--It is required so to form a homogeneous assemblage of solids of any arbitrarily given shape that each solid shall be touched by eighteen others. This problem is determinate, because the making of the three contacts _f′f_, _g′g_, _h′h_, uses up the three independent variables left at our disposal after we have first formed a homogeneous assemblage with twelve points of contact on each solid. But our manner of finding a shape for each solid which can allow the solution of the problem to be real, proves that the solution is essentially imaginary for every wholly convex shape. § 35. Pausing for a moment longer to consider afresh the geometrical problem of putting arbitrarily given equal and similar solids together to make a homogeneous assemblage of which each member is touched by eighteen others, we see immediately that it is determinate (whether it has any real solution or not), because when the shape of each body is given we have nine disposables for fixing the assemblage: six for the character of the assemblage of the corresponding points, and three for the orientation of each molecule relatively to rows of the assemblage of corresponding points. These nine disposables are determined by the condition that each body has nine pairs of contacts with others.