with Minimum Partitioned Area. 



505 



to contract. If I stop blowing before Fig. 1. 



it contracts to a point, it springs back 

 to its primitive size and shape. If I 

 blow still very gently but for a little 

 more time, the quadrilateral contracts 

 to a point, and the twelve films meeting 

 in it immediately draw out a fresh little 

 quadrilateral film similar to the former, 

 but in a plane perpendicular to its plane 

 and to the direction of the blast. Thus, 

 again and again, may the films be trans- 

 formed so as to render the little central 

 curvilinear square parallel to one or 

 other of the three pairs of square aper- 

 tures of the cubic frame. Thus we 

 see that the twelve plane films meeting 

 in the centre of the cube is a configura- 

 tion of unstable equilibrium which may 

 be fallen from in three different ways. 



6. Suppose now space to be filled 

 with equal and similar ideal rhombic 

 dodecahedrons. Draw the short diagonal of every rhombic 

 face, and fix a real wire (infinitely thin and perfectly stiff) 

 along each. This fills space with Plateau cubic frames. 

 Fix now, ideally, a very small rigid globe at each of the 

 points of space occupied by tetrahedral angles of the dodeca- 

 hedrons, and let the faces of the dodecahedrons be realized by 

 soap-films. They will be in stable equilibrium, because of the 

 little fixed globes ; and the equilibrium would be stable without 

 the rigid diagonals which we require only to help the imagina- 

 tion in what follows. Let an exceedingly small force, like 

 gravity*, act on all the films everywhere perpendicularly to 

 one set of parallel faces of the cubes. If this force is small 

 enough it will not tear away the films from the globes ; it 

 will only produce a very slight bending from the plane 

 rhombic shape of each film. Now annul the little globes. 

 The films will instantly jump (each set of twelve which meet 

 in a point) into the Plateau configuration (fig. 1), with the 

 little curve-edged square in the plane perpendicular to the 

 determining force, which may now be annulled, as we no 

 longer require it. The rigid edges of the cubes may also be 

 now annulled, as we have done with them also ; because each 

 is (as we see by symmetry) pulled with equal forces in oppo- 

 site directions, and therefore is not required for the equi- 



* To do for every point of meeting of twelve films what is done by- 

 blowing in the experiment of § 5. 



