CRYSTATJJZATION. 



271 



the excursions of the ])arts about the center of mass are on the average, 

 at a iiiven temperature and pressure, compiised within a certain ellip- 

 soid; that the dimensions <»f this ellipsoid are the same i'or all molecules 

 of the same chemical constitution, but different for molecules of differ- 

 ent kinds. 



We have now to consider how these molecnles will i)ack themselves 

 on passing from the lluid state, in which they can and do move about 

 amonost 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 a system shall be a minimnm. 

 This is the same, in the case we are considering, as saying that the 

 molecules shall be packed in such a way that the distances l)etwcen 

 their centers of mass shall on the whole be the least possible; or, that 

 as many of them as possible shall be packed into unit space. In order 

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

 first the particular case in which the axes of the ellipsoids are all 

 equal — that is, when tlie ellipsoids happen to be spheres. The prob- 

 lem is then reduced to tindiiig how to pack the greatest nuinl)cr of 

 equal spherical balls into a given si)ace. It is easy to leduce this to 

 the problem of finding how the spheres can be arranged so that each 

 one shall be touched by as many others as possil)le. En this way tlie 

 cornered si>aces between the balls, the unoccupied room, is reduced to 

 a minimum, Vou can stack balls so that each is touched by twelve 

 others, but not by more. At first sight it seems as if this might be 

 done in two wavs. 



ocoxoo 



Fio. 1. 



In the first place we may start witli a square of balls, as in Fig. 1, 

 where cacli is tonchcd by four others. We nniy then i»l;ice another 

 (shaded in the figure) so as to rest ou four, and place four niore in 



