5952 Scientific Proceedings, Royal Dublin Society. 
another, three and three in the planes of the hexagonal interfaces ; 
it will be found that the triplets of spheres thus obtained also touch 
one another in points which lie four and four in planes parallel to 
the square interfaces of the system. 
About each of the centres of the similar cells into which 
space has been partitioned, then group six equal spheres in close 
order, 7.¢., in octahedral grouping ; each octahedral grouplet to 
have its centre at the centre of the cell which contains it, and to 
have the same orientation as that of the latter, and the size of the 
spheres to be such that they touch the spheres already placed 
whose centres lie in the hexagonal interfaces. 
In the system thus formed every sphere of the central grouplets 
is in contact with four similar spheres, and four others, making 
eight contacts, and the spheres whose centres lie in the hexagonal — 
interfaces are each in contact with four of the same size, and two 
others, making six contacts... The type is that marked 13 in my ~ 
list,” and it has the generic symmetry of class 29 of Sohncke.* 
The two kinds of centres are present in the numerical proportions 
1:2. Hach kind forms a singular point system, the less numerous — 
lying on tetragonal axes, the more numerous on digonal axes. 
There are two closest-packed arrangements of the same — 
spheres, one of which arrangements is the mirror-image of the 
other. One is obtained by taking points a, fig. 11, the other by 
taking points 6 for the places of the sphere-centres of one kind. — 
Case of dodecahedral hemthedrism of the cubic system. 
Within each cell of the space partitioning of the last example 
place the centres of twelve equal spheres at equal distances from 
the centre of the cell on lines joining this centre with the middle 
points of the twelve edges in which the octahedral faces meet, and 
let the equal distances referred to and the magnitude of the spheres 
be such that the latter touch one another, and also the spheres 
1-As in the previous case, the centres of the spheres in the homogeneous structure 
thus described, will not precisely give a possible equilibrium arrangement for particles _ 
of two kinds, the necessities of statical equilibrium precluding this, but a slight modifi- 
cation of the arrangement of the sphere-centres which does not alter the type of 
symmetry will give a possible arrangement. 
* Zeitschr. fiir Kryst., 23, p. 22. 3 Ibid., 20, p. 466. 
