Bartow—A Mechanical Cause of Homogeneity of Crystals. 563 
Each smallest sphere is then in contact with three of the same 
size, and two ofeach of the larger size. Hach larger sphere of either 
size is in contact with eight small, and four of the other larger size. 
The type of homogeneous structure presented is numbered 34a, in 
my list.” The generic symmetry is that of class 23 of Sohncke’s 
list of Krystallklassen. The three kinds of centres are present 
in the proportions 1:1:4.* Allthe centres occupy singular points, 
those of the less numerous kinds lying on tetragonal axes, and all 
in planes of symmetry. 
The foregoing examples of closest-packing of one, two, or three 
different sizes of balls must suffice. To give anything like a 
complete review of the ways in which a fortuitous assemblage may 
be converted into a homogeneous assemblage as a consequence of 
closest-packing, we shall have to deal with cases of the combina- 
tion of four, five, six, &c., different sixes, and ¢o greatly multiply 
mstances in which particles of more than one kind are linked together to 
form a number of similar groups. This additional supposition often 
leads to a considerable increase of complexity which makes the 
results difficult to trace, while at the same time the number of 
possible solutions is greatly multiplied. Several, however, of the 
eases above described have been, and others can, as will subse- 
quently be pointed out,‘ be treated as those of assemblages of 
identical groups of linked particles; and it is easy to see that in 
cases of more complicated assemblages of groups of linked particles, 
as well as in the simpler cases here given, closest-packing will 
frequently lead to homogeneity of arrangement. 
Without citing otker cases, enough has been said to establish the 
general conclusion that closest-packing of an assemblage of balls 
of one, two, three, &c., different kinds will, under the conditions 
defined, very commonly lead to the production of some homogeneous 
arrangement or arrangements, the variety of form possible being 
very great indeed. All homogeneous structures whatever have 
! There will probably have to be some linking, see note 5, p. 551. 
* Zeitschr, fiir Kryst., 23, p. 45. 
5 As a fact which may have some significance, it may be noted that several bodies 
which crystallize in the symmetry referred to have a composition of the form 
AB C4. 
4 See pages 586 et seq. 
