174 



sliape is not tlie sphere (i.e. tiie form with the smallest surface) but 

 a polyhedron, is according to Gibbs and Curie owing to the following 

 circumstance. The surface energy of a surface element depends in 

 a cr3stalline substance on the orientation of tlie surface element 

 with respect to the crystalline substance, i.e. on the indices of the 

 surface elements, and this in ditferent ways for diffei'eut substances. 



If l\, k\, /l\, . . . are the capillarity constants of the differently 

 orientated bounding planes; S^, S,, /S,, . . . the corresjjonding areas of 

 the surfaces, V the volume of the crystal, then the equilibrium 

 form is characterised by the condition : 



2£ki,S/,=:i niin. for K= const (1) 



G. Wui.FF ') has derived a remarkably elegant geometrical property 

 of the equilibrium diagrams from (1), which greatly facilitates the 

 following expositions: In a figure characterised by the minimum 

 condition (1) there always exists a point IF (we will call this Wui.ff's 

 point) lying so that the distances «i, n,, . . . . of the different surfaces 

 5,, «^2 . • • fi'om W are directly proportional to the constants k^, k„ . . . 

 "i •■"^•.7i,:... — k,:k,:k,: (2) 



This theorem of Wulff's immediately furnishes a construction of 

 the equilibrium figure, if for every direction of the normal the 

 corresponding value of k has been given. Draw from an arbitrary 

 point IF of the space in all directions lines whose lengths are propor- 

 tional to the corresponding k's and apply planes normal to them 

 through their endpoints : then there remains a space in the neigh- 

 bourhood of IF, where none of these planes enters — this space is 

 the required crystalline form. It is seen here at once that surfaces 

 with a couqiaratively large value of k lie so far from IF, that they 

 cannot constitute a part of the boundaries of the crystal '). 



We derive the "law of the (small) rational indices" therefore 

 in this theory in consequence of this that the surfaces with small 

 index values in geuei'al must also possess [)articularly small capilla- 

 rity constants k. 



1) G. Wolff: Zschr. f. Kryslallogr. 34 (l'.)Ol) p. 449. The proof, which VVulff 

 had given in an imperfect form, has been impioved by Hilton afterwards : 



H. Hilton Gentralbl. f. Muier. 1901 p. 753 = Malhem. Grystallogi. (Oxford 1903) 

 p. 106. Cf. H. LiEBMANN. z. f. Kryst. 53 (1914) p. 17]-. 



') Let in the regular system e.g. the fc's of cube planes be fej, those of the 

 octahedron planes k^. It is required for the octahedron planes to occur by the 

 side of those of the cube that : 



1 k, 

 < ~ < 1/3. 



See: Curie loc. cit. and VVulff loc. cit. 



