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588 Scientific Proceedings, Royal Dublin Society. 
in equal numbers arranged in the holohedral monoclinic symmetry — 
described on page 559 have its adjacent balls all linked together, 
this assemblage can, by a symmetrical breaking of the links 
compatible with the coincidence-movements (Deckbewegungen), be 
divided into identical grouplets, each consisting of an octahedrally- 
arranged group of six of the larger balls, to each of the opposite 
triangular ends of which is linked a triad of the smaller balls, 
thus making twelve balls, six of each kind, linked together in 
each grouplet. All the six balls of each kind are similarly placed 
in the grouplet. 
A partitioning into grouplets composed of half the number of 
balls, 7.¢., three of each kind, may equally well take place con- 
sistently with the coincidence-movements (Deckbewegungen). 
4. Groups similar to one another of the composition 4B, may 
form an assemblage of type 10a, or one of type 52, and these 
groups may also be obtained from certain assemblages belonging 
to these types. 
Thus, if an assemblage composed of two kinds of balls, present 
in the numerical proportions 1 : 2, arranged in the dodecahedral 
hemihedral symmetry described on page 552, have its adjacent. 
balls all linked together, this assemblage can by a symmetrical 
breaking of the links be divided into identical grouplets of the 
composition AB,, the two similar balls being similarly situated. 
Again, an assemblage thus composed, with the trapezohedral 
tetartohedrism described on page 556, can also be divided into 
identical grouplets of the composition referred to. 
5. Groups similar to one another of the composition A,B, may 
form an assemblage of type 62a. 
Thus an assemblage composed of two kinds of balls present in 
the numerical proportions 1: 2, arranged when closest-packed, 
according to type 25a,, as described on pages 553, 554, can, without 
_ much change of the conditions of equilibrium, have its parts so 
linked and allotted as to form groups consisting of two nearest — 
large balls, about which are grouped rectangularly four smaller 
balls, the six centres of a group lying at the angles of a right — 
octahedron derived from a rectangular parallelopiped, the larger 
ones at the vertices, and all the groups being similarly orientated. 
For the modification in the arrangement to be slight in such a 
