544 Scientific Proceedings, Royal Dublin Society. 
points to form either (1) a regular octahedral space-lattice, or (2) a 
cubic space-lattice, or (8) a cubic centred space-lattice, the groups 
do not fit very closely together ; and these are the only three ways 
in which the ball-centres can be arranged so as all to occupy similar 
positions in the structure and to have cubic symmetry.” —_Closest- 
packing would appear, however, to be attained in trigonal sym- 
metry if the particles have the arrangement of a system which ~ 
has the axes of type 49 in my list® (system 21 of Sohncke), and 
possesses centres of inversion lying at the intersection of trigonal 
and digonal axes, and the generating point be so situated in a 
plane containing nearest trigonal axes of the same kind that groups 
of six points are traced from it which form the angles of regular 
octahedra. The structure is then of the type marked 49a, in my 
list. Fig. 6 will represent an arrangement of this nature, but 
the spheres will now interpenetrate in sixes, not in threes as in the 
previous case represented by this figure; and the layers will not 
now all be equidistant, the distance between consecutive layers, each 
of which contains half the centres of the same groups, being less 
than that separating layers not thus related. The symmetry dis- 
played by such a system is scalenohedral hemihedrism, being that 
of Sohncke’s class 12. Ifthe distance separating the nearest centres 
in the same group becomes equal to the distance between nearest — 
centres of different groups measured both in transverse planes of — 
particles and also between the nearest particles of succeeding — 
transverse planes, the arrangement becomes that of the very close- _ 
packed system referred to in page 534 and fig. 3. 
We might pursue the investigation for cases of centres all of — 
one kind linked to one another to form groups of more than six, 
but greater complication would then be encountered, because a 
greater number than six cannot be similarly placed with respect 
14. e., (1) To form the centres of regular rhombic dodecahedra fitted together to 
fill space; (2) to form the centres of cubes thus fitted together; or (3) to form the 
centres and solid angies of such cubes, in other words to form the centres of cubo- 
octahedra thus fitted together. 
2 Tt seems unlikely that closer packing will be attained in any homogeneous arrange- 
ment of the particles in which they have not this similarity of position. : 
3 Zeitschr. f. Kryst., 28, pp. 80 and 31. Comp. Sohncke’s Entwickelung, &e., 
p. 129. 
4 Zeitschr. f. Kryst., 238, p. 46. 
