J. W. Gibbs — EquiUbrhmi of Heterogeneous Siibstances. 467 



If the value o^ p^ is less than this, when the tensions satisfy the critical 

 relation, the point where vertices of the masses A, B, C, D meet is 

 stable with respect to the formation of any mass of the nature of E. 

 But if the value of p-^ is greater, either the masses A, B, C, D cannot 

 meet at a point in equilibrium, or the equilibrium will be at least 

 practically unstable. 



When the tensions of the new surfaces are too small to satisfy the 

 critical relation with the other tensions, these surfaces will be con- 

 vex toward E ; when their tensions are too great for that relation, 

 the surfaces will be concave toward E. In the first case, Wy is 

 negative, and the equilibrium of the five masses A, B, C, D, E 

 is stable, but the equilibrium of the four masses A, B, C, D meeting 

 at a point is impossible or at least practically unstable. This is sub- 

 ject to the limitation that when p^ is sufficiently small the mass E 

 which will form will be so small that it may be neglected. This will 

 only be the case when p^ is smaller — in general considerably smaller — 

 than the second number of (642). In the second case, the equilibrium 

 of the five masses A, B, C, D, E will be unstable, but the equilibrium 

 of the four masses A, B, C, D will be stable unless v-e (calculated for 

 the case of the five masses) is of insensible magnitude. This will 

 only be the case when p)^ is greater — in general considerably greater — 

 than the second member of (642). 



Liquid Films. 



Wlien a fluid exists in the form of a thin film between other fluids, 

 the great inequality of its extension in different directions will give 

 rise to certain peculiar properties, even when its thickness is sufficient 

 for its interior to have the properties of matter in mass. The fre- 

 quent occurrence of such films, and the remarkable properties which 

 they exhibit, entitle them to particular consideration. To fix our 

 ideas, we shall suppose that the film is liquid and that the contiguous 

 fluids are gaseous. The reader will observe our results are not 

 dependent, so far as their general character is concerned, upon this 

 supposition. 



Let us imagine the film to be divided by surfaces perpendicular to 

 its sides into small portions of which all the dimensions are of the 

 same order of magnitude as the thickness of the film, — such portions 

 to be called elements of the film,, — it is evident that far less time will 

 in general be required for the attainment of approximate equilibrium 

 between the different parts of any such element and the other fluids 

 which are immediately contiguous, than for the attainment of equi- 



