534 Scientific Proceedings, Royal Dublin Society. 
of different ways. There are two, and two only, which give 
homogeneity of structure,’ viz. :— 
(a.) An arrangement in which the sphere centres occupy the 
centres of a symmetrically selected half of the cubes of a.cubic par- 
titioning of space. This is shown in figs. 1 and 2,’ and is of ‘the 
type marked 8a, in the lists of types of homogeneous structure 
given by the writer,? and has the generic symmetry of class 
28 in Sohncke’s list of Krystallklassen.* The centres form a 
singular point-system,° each of them being the intersection of 
tetragonal, trigonal, and digonal axes and of the planes of 
symmetry. 
(6.) An arrangement in which the spheres of alternate layers 
are directly over one another, the pro- 
jection of the system being shown in 
fig. 3. The sphere centres then exem- 
plify that particular case of Sohncke’s 
system 02, which is obtained when the 
generating point lies at the point of 
intersection of one of the trigonal and 
one of the digonal axes of the system, 
and the distance separating successive 
layers of the system is such that each 
point is equidistant from twelve nearest points. Such a system 
possesses planes of symmetry in which the points lie, and also 
centres of inversion, and is a singular point-system. The type of 
homogeneous structure to which it belongs is that marked 24a,,° 
the ‘“‘ doppelte Systeme,’ of which are of the kind numbered 87a 
1 See note 1, p. 531. 
2 Fig. 2 has one corner of a cubic group truncated to show the triangular close- 
packed arrangement of the spheres in planes perpendicular to the cube diagonals. 
3 Zeitschr. fiir Kryst. 23, p. 44. The references given here, and in subsequent 
examples to the lists contained in the author’s former works on homogeneous struc- 
tures, and to Sohncke’s list of the thirty-two classes of crystal symmetry are supplied 
for the convenience of those who desire to examine particular cases closely ; it is scarcely 
necessary for the purposes of the general argument that the reader should look them up, 
4 Ibid., 20, p. 466. 
5 Ibid., 23, p. 60. 
§ Zeitschr. fiir Kryst. 23, p. 45. Compare Sohncke, Entwickelung einer Theorie 
der Krystallstruktur, p. 115; also Natwre, Dec. 20, 1883, vol. xxix., p. 186. 
