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LXTII. On the Division of Space with Minimum Partitioned 

 Area. By Sir William Thomson*. 



1. rTlHIS problem is solved in foam, and the solution is 

 J- interestingly seen in the multitude of film-enclosed 

 cells obtained by blowing air through a tube into the middle 

 of a soap-solution in a large open vessel. I have been led to 

 it by endeavours to understand, and to illustrate, Green's 

 theory of " extraneous pressure " which gives, for light tra- 

 versing a crystal, Fresnel's wave-surface, with Fresnel's sup- 

 position (strongly supported as it is by Stokes and Rayleigh) 

 of velocity of propagation dependent, not on the distortion- 

 normal, but on the line of vibration. It has been admirably 

 illustrated, and some elements towards its solution beautifully 

 realized in a manner convenient for study and instruction, 

 by Plateau, in the first volume of his Statique des Liquides 

 soumis aux seules Forces MoUcutaires. 



2. The general mathematical solution, as is well known, is 

 that every interface between cells must have constant cur- 

 vaturef throughout, and that where three or more interfaces 

 meet in a curve or straight line their tangent-planes through 

 any point of the line of meeting intersect at angles such that 

 equal forces in these planes, perpendicular to their line of 

 intersection, balance. The minimax problem would allow any 

 number of interfaces to meet in a line ; but for a pure minimum 

 it is obvious that not more than three can meet in a line, and 

 that therefore, in the realization by the soap -film, the equi- 

 librium is necessarily unstable if four or more surfaces meet 

 in a line. This theoretical conclusion is amply confirmed by 

 observation, as we see at every intersection of films, whether 

 inter facial in the interior of groups of soap-bubbles, large or 

 small, or at the outer bounding-surface of a group, never more 

 than three films, but, wherever there is intersection, always 

 just three films, meeting in a line. The theoretical conclusion 

 as to the angles for stable equilibrium (or pure minimum 

 solution of the mathematical problem) therefore becomes, 

 simply, that every angle of meeting of film-surfaces is exactly 

 120 / 



3. The rhombic dodecahedron is a polyhedron of plane sides 

 between which every angle of meeting is 120°; and space can 



* Communicated by the Author. 



t By " curvature " of a surface I mean sum of curvatures in mutually 

 perpendicular normal sections at any point ; not Gauss's " curvatura 

 integra," which is the product of the curvature in the two " principal 

 normal sections/' or sections of greatest and least curvature. (See 

 Thomson and Tait's ' Natural Philosophy,' part i. §§ 130, 136.) 



