540 Scientific Proceedings, Royal Dublin Society. 
projections of alternate layers are identical, so that two projections 
of succeeding layers suffice. The positions of the centres are all 
identical; they are singular points, and the digonal axes on which 
they lie are lines of intersection of those planes of symmetry which 
intersect trigonal and hexagonal axes. 
Suppose next— 
(3.) That the distances between the three ball centres forming 
a group are almost as great as the distances between the nearest 
centres in different groups when equilibrium is reached, so as very 
nearly to permit of the centres taking the arrangement of the 
sphere centres in figs. 1 & 2. 
If the distances referred to differ only infinitesimally, we shall 
get an unhomogeneous arrangement very closely resembling in 
structure the closest-packed arrange- 
ment referred to, but lacking its definite 
polarity. If they are not infinitesimal 
it would appear that the conditions may 
be such as to give the closest-packing 
in an arrangement depicted in fig. 6, in 
which the three centres grouped around 
a hexagonal axis and lying in the 
same transverse layer are rather nearer 
together than to centres in other groups; Fig. 6. 
as before the spheres interpenetrate in 
threes. The system has the axes of system 52 of Sohncke (Type 
24 in my list) with centres of inversion on the hexagonal axes and 
situated midway between the layers of points. Its structure is 
therefore of the type marked 24a, in my list,’ and has the generic 
symmetry of class 9 in Sohncke’s list.2 The positions of the 
centres are all identical; they are singular points lying on those 
of the digonal axes which intersect nearest hexagonal axes and 
also on lines of intersection of planes of symmetry. 
Take next a case in which the centres, while all of one 
kind, are similarly linked to one another to form groups of 
four. 
1 Zeitschr. f. Kryst., 23, p. 45. The ‘‘doppeltes System” is No. 88a of Fedorow 
(see Zeitschr. f. Kryst., 24, viii., Taf. vi.). 
2 Zeitschr. f. Kryst., 20, p. 460. 
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