828 Professor William J. Pope [Ap^'il l-'^- 



now seen that the even rate of expansion from each point of the 

 original point-system which gives rise to the closely packed stack of 

 rhombic dodecahedra, symbolises an even radiation in all directions 

 of the forces of which the atom is the centre of emanation. On 

 applying the same operation of expansion to the spheres present in 

 hexagonal closest-packing, each becomes converted into a dodeca- 

 hedron, although of symmetry different from that of the rhombic 

 dodecahedron. In each of the two cases the system exhibits the 

 important property that, with a given density of distribution of the 

 centres, a maximum distance prevails between nearest centres ; these 

 two systems thus represent the equilibrium arrangements of the postu- 

 lated forces of repulsion exerted between near centres, the repulsions 

 between more distant ones being neglected. 



It will be sufficiently evident from what has been said that the 

 function of the spherical surfaces in the closest-packed assemblages of 

 spheres, as representing crystal structures, is merely a geometrical 

 one ; these surfaces are employed only as so much scaffolding by the 

 aid of which may be derived arrangements exhibiting a maximum 

 number of equal distances between neighbouring centres, and no phy- 

 sical distinction is to be made betAveen portions of space lying within 

 the spheres and portions forming part of the interstices between them. 

 Insistence on this point is necessary, because many investigators have 

 made use, quite illegitimately, of spheres for the representation of 

 atomic domains, piling the spheres together in what they have termed 

 open packing ; this term seems to imply that some physical difference 

 can subsist between the portions of space lying within the spheres and 

 those lying without. The one kind of space is apparently regarded 

 as susceptible to atomic influence in some sense not exhibited by the 

 other. To state this view in any definite manner probably suffices to 

 demonstrate its superficiality : the question of ascertaining what pro- 

 portion of the total space is available for atomic occupation by the 

 use of assemblages of spheres does not arise because the spheres used 

 are solely the geometrical instruments for producing equality amongst 

 the atomic distances, and so determining the prevailing equilibrium 

 conditions. 



So far as the enquiry has been carried, it would seem that the 

 elements should crystallise either in the cubic or the hexagonal 

 system, and that in the latter case corresponding dimensions in the 

 horizontal and vertical directions should be in the ratio of a :c = 

 1 : 0*8165. The facts are summarised in Table I. 



Of the elements which have been crystallographically examined, 

 50 per cent, are cubic ; their crystal structure is simulated by the 

 cubic closest-packed assemblage of equal spheres. Another 35 per 

 cent, belong to the hexagonal system, and that these are correctly 

 represented by the hexagonal closest-packed assemblage of equal 

 spheres is indicated by the fact that foi' the hexagonal elements, the 

 ratio of corresponding dimensions in the horizontal and vertical direc- 



