June i8, 1891] 



NATURE 



»57 



Again, if we start with such a triangle, as in Fig. 3, where 

 each ball is touched by six others, we can place one ball 

 — the shaded one — so as to rest on three others, and can 

 then place six more round it and touching it, as in- 

 dicated by the dotted circles. In three of the triangular 

 holes between the shaded ball and the dotted balls touch- 

 ing it we can place three more, so as to touch the shaded 

 ball — again twelve touching it in all. If we complete 



the pile, we shall get the triangular pyramid represented 

 by Fig. 4, where each of the three sides is a right-angled 

 triangle, while the base is an equilateral triangle. It will 

 be seen that in the faces of this pyramid each ball 

 (except those outside) is touched by four others. In 

 fact, the arrangement in these faces is the same as in the 

 base of the former pyramid ; and the two arrangements 

 are really identical in the interior, only one has to be 



turned over in order to bring it into parallelism with the 

 other. Fig. 2 represents half a regular octahedron ; 

 Fig. 4 the corner of a cube. Ellipsoids, if they are all 

 equal and similar to one another, can be packed in pre- 

 cisely the same way, so that each is touched by twelve 

 others, provided their axes are kept parallel to each other 

 —that is, if they are all oriented alike. This, then, by the 

 laws of energy, will be the arrangement which the mole- 



NO. I I 29, VOL. 44] 



cules will assume, in consequence of mutual attraction, iip 

 passing from a fluid to a solid state. 



Next, let us see how the packing of the molecules wilt 

 affect the external form. And here I bring in the surface- 

 tension. We are familiar with the effects of this force in 

 the case of liquids, and if we adopt the usually received 

 theory of it, we must have a surface-tension at the 

 boundary of a solid, as well as at the surface of a liquid. 

 I know of no actual measures of the surface-tension of 

 solids ; but Quincke has given us the surface-tensions 

 of a number of substances at temperatures near their 



I points of solidification, 

 follows :— 



Platinum 



Gold 



Zinc 



Tin 



Mercury 



Lead 



Silver 



Bismuth 



Potassium 



Sodium 



in dynes per lineal centimetre, as^ 



1658 

 9«3 

 860 

 587 

 577 

 448 



Antimony 



Borax 



Sodium carbonate 

 Sodium chloride 



Water 



Selenium 



419 Sulphur 



382 Phosphorus ... 



364 Wax 



253 



244 



212 



206 



114 

 86-2 

 704 

 41-3 

 411 

 33 "4 



The surface-tensions of most of the solids are probably 

 greater than these, for the surface-tension generally 



Fig. 4. 



diminishes with increase of temperature; and you see 

 that they amount to very considerable forces. We have 

 to do, then, with an agency which we cannot neglect. Iiv 

 all these cases the tension measured is at a surface bounded 

 by air, and is such as tends to contract the surface. We 

 have, then, at the boundary between a crystallizing solid 

 and the fluid, be it gas or liquid, out of which it is solidi- 

 fying, a certain amount of potential energy ; and by the 

 laws of energy the condition of equihbrium is, that this- 

 potential energy shall be a minimum. The accepted 

 theory of surface-tension is that it arises from the mutual 



