536 Scientific Proceedings, Royal Dublin Society. 
If, for example, the similar centres are similarly linked together 
two and two, closest-packing gives different arrangements accord- 
ing to the distance apart of the two centres of a pair as compared 
with the distances separating nearest centres in different pairs 
when equilibrium is reached. The consideration of a few particular 
cases renders this clear. : 
Thus suppose :— 
(a) That the distance between the two centres of a pair is small’ 
compared with the distances between those of the nearest pairs 
when equilibrium is reached, so that the balls of the pair inter- 
penetrate one another very considerably. 
When this is so the arrangement which gives closest-packing 
will still, very approximately, be that of the simpler case given 
above, except that now the points midway between the two centres 
in each group or pair, instead of the centres themselves, will 
have the arrangement of the sphere centres in the assemblage 
depicting a closest-packed arrangement of unlinked spheres of a 
single kind. 
The assemblage will not, however, be homogeneous unless the 
relative orientation of the pairs conforms to the definition of 
homogeneousness above referred to. ‘The only relative orientation 
which conforms to this definition, and also has the symmetry of 
the regular form, is that in which the centres all lie on trigonal 
axes situated as in type 1 of my list? (System 58 of Sohncke), the 
line joining the two centres of a pair coinciding everywhere with 
such an axis. If the pairs have this relative orientation, a homo- 
geneous structure is presented which has the axes and coincidence- 
movements (Deckbewegungen) of type 1, and possesses in addition 
centres of symmetry (of inversion) lying at the cube centres of the 
space partitioning which I have employed to generate this system.® 
It is, therefore, a structure of the type numbered La, in my list,* and 
displays the generic symmetry of class 31 in Sohncke’s list of Krys- — 
tallklassen.? The two centres of a pair are equidistant from the 
centre of the cube in which they lie, and they are found on the 
single trigonal axis of this cube; their positions are not identically 
1 Not infinitesimal. 2 Zeitschr. fir Kryst., 23, p. 6. 
3 Ibid., 23, p. 7- 4 Tbid., p. 44. 5 Tbid., 20, p. 467. 
