Bartow—A Mechanical Cause of Homogeneity of Crystals. 5833 
disposition of the ball centres which gives closest-packing, is one in 
which each ball is in contact with twelve others. For if a number of 
equal spheres be stacked together, twelve is the greatest number 
of them which can be in contact with any given sphere, and a 
closest-packed arrangement is realized by the sphere-centres of a 
stack consisting of plane layers triangularly arranged in which 
each sphere is in contact with six others, and the succeeding layers 
are so disposed that every sphere is also in contact with three others 
of the layer above, and three of the layer below it. 
But since there are two different positions in which a second 
plane layer can be deposited to fulfil this condition, it is evident 
that twelve contacts for each sphere can be attained in a variety 
1 Kven this simplest of all cases of closest-packing has not, so far as I am aware, 
ever been exhaustively treated, or its various possibilities expressed algebraically. 
(See Lord Kelvin on the arrangements here referred to in ‘‘ Molecular Constitution of 
Matter,’’ by Sir William Thomson, Proc. Royal Soc. of Edinburgh, vol. xvi., p. 712, 
note *and p. 715.) As those who have employed the conception of closest-packing of 
similar bodies have contented themselves with a treatment which is comparatively 
rudimentary and not exhaustive, considerable difficulty must be looked for in any 
attempt to deal exhaustively with the closest-packing of bodies of more than one kind. 
I am, however, not without hope that, when the importance of the subject is realized, 
experts in analytical methods may he found who will indicate precise ways of arriving 
at some of the closest-packed arrangements described in the succeeding pages. I 
anticipate that should some of these arrangements prove not to be the closest-packed 
possible, the arrangements which supplant them will in all, or nearly all, cases be found 
to be homogeneous, and will therefore serve the purposes of the argument equally 
well. 
I may say that my general principle for getting closest-packing of the spheres is to 
produce a maximum number of contacts, so as to diminish, as far as possible, the 
amount of interstitial space. 
