538 Scientific Proceedings, Royal Dublin Society. 
shown by fig. 3.1. In such a grouping, the centres linked are 
in different layers, and the positions of the two ball centres of the 
pairs are identical;? they are singular points in planes of symmetry. 
The two projections of alternate layers are shown in the figure. 
Layers between which the linking obtains will be rather closer 
together than other succeeding layers.* The relative position of 
two linked balls is indicated by connecting with dotted lines the 
projections of the spheres whose centres pair together. 
Finally suppose— 
(c) That the distance between two centres of a pair is slightly 
but not much less than in the last case. | 
It is then conceivable that the closest-packing will be reached 
in a homogeneous arrangement belonging to the regular system 
whose axes are those of type 5 in my list* (system 62 of Sohncke), 
and which is obtained by placing a point on a trigonal axis very near 
either to one of the cube centres, or to one of the cube angles of the 
space-partitioning employed to generate this system,’ and generating 
a point-system by carrying out the coincidence-movements (Deck- 
bewegungen) of the samesystem. ‘The ball centres to be regarded 
as linked together will be those lying nearest to one another on the 
same trigonal axis. The structure obtained will be of the type 
numbered 5 in my list, and display the generic symmetry of class 29 
in Sohncke’s list. The positions of the centres are all identical, and 
they are singular points lying on trigonal axes. 
Where, on stable equilibrium being reached, the distance 
between two centres forming a pair bears some other relation to 
the distances between the pairs than those prevailing in the three 
cases referred to, the closest-packed arrangement, towards which 
the assemblage continually approximates, will be in some cases. 
homogeneous, in other cases probably unhomogeneous. To be 
homogeneous it must, as we know, possess the coincidence-move- 
ments of some one of the 65 systems of Sohncke, and will generally 
be a specialized system of very high symmetry.” 
1 See page 584. 2 4.e., not enantiomorphous. 
3 If the layers were all equidistant, the symmetry would, ignoring the linking, 
belong to one of the hexagonal types. 
4 Zeischr. fiir Kryst., 23, pp. 6and 12. ° [did., 23, pp. 7and 12. ° bid., 20, p. 466. 
7 See ‘‘ Ueber die geometrischen Higenschaften &c.’’ Zeitschr. fur Kryst., 23, p. 1. 
