508 Sir W. Thomson on the Division of Space 



please annul, the difference of pull per unit area on the three 

 pairs of sides of the cage. The respective shrinkage-ratio 

 and extension-ratio, to exactly equalize the pulls per unit area 

 on the three principal planes, (and therefore on all planes), are 



2~~ 5 , 2 r , 2% as is easily seen from what follows. 



9. While the equalization of pulls in the three principal 

 directions is thus produced, work is done by the film on the 

 moving wire- work of the cage, and the total area of film is 

 diminished by an amount equal to W/T, if W denote the 

 whole work done, and T the pull of the film per unit breadth. 

 The change of shape of the cage being supposed to be per- 

 formed infinitely slowly, so that the film is always in equi- 

 librium throughout, the total area is at each instant a mini- 

 mum, subject to the conditions 



(1) That the volume of each cell is the given amount; 



(2) That every part of the wire has area edged by it ; and 



(3) That no portion of area has any free edge. 



10. Consider now the figure of the cell (still of course a 

 tetrakaidecahedron) when the pulls in the three principal 

 directions are equalized, as described in § 8. It must be 

 perfectly isotropic in respect to these three directions. Hence 

 the pair of small quadrilaterals must have become enlarged 

 to equality with the two pairs of large ones, which must have 

 become smaller in the deformational process described in 

 § 8. Of each hexagon three edges coincide with edges of 

 quadrilateral faces of one cell ; and each of the three others 

 coincides with edges of three of the quadrilaterals of one of 

 the contiguous cells. Hence the 36 edges of the isotropic 

 tetrakaidecahedron are equal and similar plane arcs ; each of 

 course symmetrical about its middle point. Every angle of 

 meeting of edges is essentially 109° 28' (to make trihedral 

 angles between tangent planes of the films meeting at 120°). 

 Symmetry shows that the quadrilaterals are still plane 

 figures ; and therefore, as each angle of each of them is 

 109° 28', the change of direction from end to end of each arc- 

 edge is 19° 28'. Hence each would be simply a circular arc 

 of 19° 28', if its curvature were equal throughout ; and it 

 seems from the complete mathematical investigation of §§ 16, 

 17, 18 below, that it is nearly so, but not exactly so even to a 

 first approximation. 



Of the three films which meet in each edge, in three 

 adjacent cells, one is quadrilateral and two are hexagonal. 



11. By symmetry we see that there are three straight lines 

 in each (non-plane) hexagonal film, being its three long dia- 

 gonals ; and that these three lines, and therefore the six 

 angular points of the hexagon, are all in one plane. The arcs 



