1891.] on Crystallisation. 377 



are, on tlie average, at a given temperature and pressure, comprised 

 within a certain ellipsoid : that the dimensions of this ellipsoid 

 are the same for all molecules of the same chemical constitution, but 

 different for different kinds of molecules. 



We have next to consider how these molecules will pack them- 

 selves in passing from the fluid state, in which they can and do move 

 about amongst themselves, into the solid state, in which they have no 

 sensible freedom. If they attract one another according to any law, 

 and for my purpose gravity will suffice, then the laws of energy 

 require that for stable equilibrium the potential energy of the system 

 shall be a minimum. This is the same, in the case we are considering, 

 as saying that the molecules shall be packed in such a way that the 

 distance between their centres of mass shall be the least possible, or 

 as many of them as possible be packed into a given space. 



In order to see how this packing will take place, it will be easiest 

 to consider the case in which the axes of the ellipsoids are all equal 

 — that is, when the ellipsoids happen to be spheres. The problem 

 is then reduced to finding how to pack the greatest number of equal 

 spherical balls into a given space. It is easy to reduce this problem 

 to that of finding how the spheres can be arranged so that each sphere 

 shall be touched by as many as possible of its neighbours. In this 

 way the cornered spaces between the spheres, the spaces not occupied, 

 are reduced to a minimum. Now, you can arrange balls so that each 

 is touched by twelve others, but not by more than twelve. This, 

 then, will be the arrangement which the molecules will naturally 

 assume. 



Fig. 1. 



We may do this apparently in two ways. We may begin with 

 arranging balls on a flat surface so that each is touched by six others, 

 as in Fig. 1. We may then place a ball so that it rests on three, a, 6, c, 



