of Homogeneous Partitioning s of Space. 



of the cube elongated parallel to four 



91 



the case of the cube elongated parallel to four of its 

 edoes and transformed into a square prism. The inflexion 

 characteristic for this curve is equally apparent in the four 



Fig. 5. 



L/Lnr 



others, and occurs, so far as can be judged, at the same 

 abscissa. What the geometrical meaning of this value may 

 be is difficult to see in view of the difference in type between 

 the various polyhedra which have been considered. 



The problem which gave rise to the present investigation 

 is probably the first one in which deformations of homo- 

 geneous partitionings of space had to be taken into con- 

 sideration, while the well-known applications of the theory, 

 as made by Osborne Reynolds, Pope and Barlow, and others, 

 did not call for an examination of the changes produced in 

 the partitioning by stresses. Such an examination cannot, 

 however, be avoided in those instances in which a polyhedral 

 structure and the existence of an inter-polyhedral layer 

 with special properties are postulated to account for the 

 peculiarities of systems under stress, e. g. the theory of an 

 intercrystalline cement as an important factor in the tensile 

 strength of metals, &c. The results here given, which not- 

 withstanding the somewhat special assumptions seem to point 

 to a uniform behaviour of homogeneous partitionings, may 

 therefore be useful as providing a direct criterion of the 

 validity of speculations which postulate such structures. 



November 11. 1916. ' 



