530 Scientific Proceedings, Royal Dublin Society. 
rate from those of another material, and (2) it is postulated that 
ball can be attached to ball bya practically inextensible tie reaching 
from centre to centre through the place of contact, so that when 
tied in this way, the centres of two balls cannot get further and 
further apart under change of conditions, and thus the balls have, 
so to speak, to inter-penetrate between tied centres if they expand. 
The last-named postulate is intended to imitate to some extent 
the different kinds of aggregation of the ultimate parts of 
matter.’ 
The simple materials thus provided,’ the different balls 
of different materials,®> with their power of expansion and 
contraction under change of conditions, and their faculty 
of attachment to form groups, are now to be put together in 
various ways which fulfil the condition of closest-packing, this principle 
being the foundation stone of the present inquiry. The arrange-_ 
ments obtained will be compared with various phenomena of 
erystallization,* chemical isomerism, chemical combination, and 
diffusion. 
The work may be regarded as supplementary to the geometrical 
work on the nature of homogeneity which the author has already 
published, by which he has shown that every homogeneous struc- 
ture, whatever its nature, displays one or other of the thirty-two 
kinds of crystal symmetry.° 
The various effects producible will be found to range themselves 
under seven heads, viz. :— | 
I.—Symmetrical arranging of parts converting a fortuitous 
assemblage into a homogeneous assemblage and subsequent preser- 
vation of the homogeneity by the application of the ties, an effect 
1 Where the linking thus defined is sufficiently complete to prevent any free move- 
ment of small parts of the assemblage with respect to one another, the assemblage will 
in the sequel, be said to be solid. 
2 For a more precise statement of the concepts employed, see Appendix. 
5 Sometimes it will be convenient to think of an artificial system as consisting of 
mutually-repellent particles instead of balls of different kinds (see above). 
4 A reference to some of the conclusions as to crystals reached in this memoir was 
made by the author at the Cardiff Meeting of the British Association, 1891. Diagrams — 
showing some of the kinds of symmetry dealt with were published in Nature, 1883, 
vol. 29, pp. 186 and 205. 
5 See Mineralogical Magazine, vol. xi., p. 119. 
