Bartow—A Dechanical Cause of Homogeneity of Crystals. 549 
them all, there will, when equilibrium is reached, be an excess 
containing one kind of ball less than is found in this combination, 
and that closest-packing applied to this excess will cause the balls 
composing it to take up the relative positions in which they 
occupy the least space possible! We see, therefore, that the 
numerical proportions in which different kinds of balls are present 
will determine how far uniformity of arrangement shall extend 
throughout the entire assemblage. Where there are but two 
kinds of balls, as in the cases referred to above, the combination 
ultimately produced will exhaust the stock of one kind, and the 
kind which is present in excess will be arranged just as the sphere- 
centres are arranged in the closest-packed system above referred to.” 
To proceed to other cases of equilibrium of two kinds of balls. 
Another case of holohedral cubic symmetry. 
The two sizes may be so related that the disposition of the 
centres when the most stable equilibrium is 
reached is that of the centres of spheres of 
two sizes, where those of one size occupy 
the centres of all the cubes of a cubic par- 
titioning of space, those of the other all the 
cube angles (fig. 9). In this case all the 
balls will be operative.? The type of homo- 
geneous structure to which such an assem- 
blage belongs is that numbered 7a, in my 
list.‘ Each kind of centre forms a singular point system,” centres of 
Fig. 9. 
' As to the steps by which this ideally-perfect condition of most stable equilibrium 
will conceivably be reached in the case of an assemblage of mutually-repellent 
particles of two different kinds, patches of such a symmetrical combination as gives 
closest-packing will probably first appear at all places where the different kinds are in 
juxtaposition, then the two kinds will interpenetrate each other, and the patches of 
the compound assemblage formed will extend and combine till an arrangement is 
reached in which all of one kind of particle is in combination with the needful 
proportion of the other in a continuous mass as symmetrically arranged as possible. 
The remainder of the assemblage will consist entirely of the kind of particle which is 
present in excess arranged in one of the closest-packed forms for the uncombined state, 
probably in that depicted in figs. land2. There will, of course, be some irregularity and 
continual fluctuation at all boundaries. It is needless to add that, if change of state 
takes place before the ideally-perfect condition is reached, the process described will be 
only partially carried out. * See figs. 1 and 2. 
3 See p. 547. 4 Zeitschr. fiir Kryst., 28, p. 44. 5 Ibid., p. 60. 
