Bartow—A Mechanical Cause of Homogeneity of Crystals. 651 
both,’ the positions occupied by single balls or by the centres of highly 
symmetrical groups of balls in one assemblage are occupied in the other 
by groups whose type of symmetry is low enough to impose a lower 
symmetry on the latter assemblage. 
Suppose, for example, to take a very simple case, that in two 
homogeneous isomorphous assemblages the singular points have 
the same relative arrangement, both containing balls of a certain 
kind situated at the centres of equal regular hexagonal right 
prisms of a particular pattern filling space in the most symmetrical 
manner possible, but that while in the one assemblage the remain- 
ing balls consist of triads of another kind of ball with their centre- 
points at the points in which six prism corners meet, in the other 
assemblage they consist of right tetragonal groups each composed 
of four balls of a third kind with the group centres placed in the 
same way as those of the triads.’ 
It is then manifest that if the principle of closest-packing 
requires the groups to be orientated in the most symmetrical way 
possible, the inclinations of the principal planes of points to one 
another will be the same in the two assemblages, which will be 
capable of intercalation in any proportions, but that the symmetry 
of the one containing the triads will be of a higher class than that 
of the other, the former having the symmetry of type 25a, the 
latter that of type 23b,, in my tables of homogeneous structures.* 
The isomorphism displayed by two allied assemblages will 
commonly not be perfect if the balls which are unlike in them are 
not absolutely inoperative in the sense above defined; and whether 
they are so or not, if the balls common to both do not behave 
identically under the same external conditions, but are of different 
magnitude* according to the nature of the balls with which they 
are associated in the two cases, it is very unlikely that the equality 
of corresponding angles will, except in the regular system, be 
1 The ratios referred to appear to be the same as what are called by Tutton “ dis- 
tance ratios,’’ by Muthmann ‘‘ topic ”’ or “ topical axial ratios.’’ See Zeitschr. fiir 
Kryst., 22, p. 497, and Journ. Chem. Soc., vol. lxy., p. 628. We see that for their 
determination it is mot necessary to partition homogeneous structures in any way. 
2 Neither arrangement is capable of being so linked as to form groups of a single 
kind without deteriorating the symmetry. It would, of course, be easy to instance 
others which are capable. 
3 Zeitschr. f. Kryst, 23, pp. 45 and 52. * See page 529. 
