528 Scientific Proceedings, Royal Dublin Society. 
and distinguish a man from his horse, while at the same time 
generally, in a sense, regarding them as one." 
Perhaps the simplest conceivable kind of closest-packing whial 
gives this diversity in unity of the elements of the structure, is that 
of a large number of spheres of two, three, or more different sizes ; 
and, as will be seen in the following pages, the variety of types of 
symmetry obtainable in this way, when a great number of different 
ratios between the sizes of the spheres are taken, is comparable to 
the variety presented by crystals. 
When further we find that numerous other facts concerning 
crystals are paralleled by the properties of closest-packed sym- 
metrical assemblages composed of balls of different sizes, we may 
be tempted to conclude that, in cases where closest-packing of balls 
all of the same size does not suflice, the real state of things is 
pictured in outline by the simple kind of closest-packing just 
referred to, and if this be so we shall conceive the ‘spheres of 
influence’ of the atomic properties or movements, whatever these 
may be, to be strictly spherical around certain centres, so that 
polarity in a crystal is traceable to the disposition and not to the 
forms of the spheres of influence of its ultimate parts. 
This is, perhaps, not unlikely, but it is proper to remark that 
the existence of the parallelism alluded to by no means establishes 
the propriety of using spherical balls in all cases. For every 
crystal not of the cubic system can be conceived to be deformed in 
such a way as not to alter the type of symmetry of its parts or 
properties, and since corresponding deformation of an assemblage 
of balls representing its symmetry would alter the spheres to 
spheroids or ellipsoids, we see that while the latter forms are not 
necessarily indispensable for the portrayal of the lower types of 
symmetry, they will always do as well as spheres so far as the 
geometry is concerned. 
An objection to the simple kind of closest-packing in question 
1Tf the definition of homogeneity of structure suggested by the author (see 
Mineralogical Magazine, vol. xi., p. 120) is adopted, there are types of homogeneous 
arrangement whose representation would require the units of the troop to be no¢ all 
similarly orientated, and some which would involve the military inconvenience of each 
man being related similarly to more than one horse, and each horse related similarly 
to more than one man. 
