84 Mr. E. Hatschek on some simple Deformations 



to exist in the high percentage emulsions, and is known to 

 persist for years. This is, no doubt, due to the very low 

 interfacial tension, which is a necessary condition for the 

 production of such emulsions ; as gels ex liijpotliesi consist 

 of two phases differing only in concentration, but not in 

 composition, the interfacial tension is likely to be lower even 

 than that between the soap solutions and mineral oils of 

 Pickering's emulsions, so that the continued existence of a 

 polyhedral structure offers no difficulty. 



As regards the geometrical character of the interface, we 

 shall hardly be mistaken in assuming that it conforms to 

 some type of homogeneous partitioning of space, and that its 

 elements are either twelve- or fourteen-faced polyhedra. If 

 we determine the increase of surface produced by deforming 

 such a structure and plot the first differential coefficients of 

 these increases as ordinates against the elongations, we 

 obtain, as already explained, the stress-elongation curves. 

 As the physical results of the investigation have been pub- 

 lished in a short paper *, it is only necessary to state here 

 that these theoretical curves show very marked disagreement 

 with the experimental stress-elongation curves. 



To extend the geometrical investigation to all cases, we 

 shall consider, in addition to dodecahedral and tetrakai- 

 decahedral partitionings, the hexahedral partitioning. If we 

 confine ourselves to polyhedra with straight edges and plane 

 faces, the minimum case, i. e. the polyhedron with the 

 minimum surface, will be : for the hexahedral partitioning 

 the cube, for the dodecahedral type the regular rhombo- 

 dodecahedron and for the tetrakaidecahedral type the equi- 

 lateral cubo-octahedron. As regards the deformation, we 

 are only limited by the condition of continuity that the 

 polyhedra obtained from the minimum types must again fill 

 space continuously. This condition is, of course, satisfied by 

 the following types of deformation, which we will consider 

 in detail : — 



(1) Hexahedral partitioning. — (a) The cube is deformed 

 by a stress parallel to four of its edges and transformed into 

 a square prism having the same volume. Although this case 

 appears highly special, it will be found to be in complete 

 functional agreement with all the others. 



(b) The cube is deformed by a stress acting in the 

 direction of its space diagonal and all edges remain equal, so 

 that it is transformed into a rhombohedron (fig. 1, a & b). 



* Trans. Faraday Society, vol. xh\ pt. 1 (1916). 



