262 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



of insensible magnitude. {The diminution of the radii with increasing 

 values of p p A is indicated by equation (565).} Hence, no mass of 

 phase C will be formed until one of these limits is reached. Although 

 the demonstration relates to a plane surface between A and B, the 

 result must be applicable whenever the radii of curvature have a 

 sensible magnitude, since the effect of such curvature may be dis- 

 regarded when the lentiform mass is sufficiently small. 



The equilibrium of the lentiform mass of phase C is easily proved 

 to be unstable, so that the quantity W affords a kind of measure of 

 the stability of plane surfaces of contact of the phases A and B.* 



Essentially the same principles apply to the more general problem 

 in which the phases A and B have moderately different pressures, so 

 that their surfaces of contact must be curved, but the radii of curva- 

 ture have a sensible magnitude. 



In order that a thin film of the phase C may be in equilibrium 

 between masses of the phases A and B, the following equations must 

 be satisfied: , 



where c^ and c 2 denote the principal curvatures of the film, the 

 centers of positive curvature lying in the mass having the phase A. 

 Eliminating Cj + Cg, we have 



(PA. -PC) = <TAC (Po -Pv)> 



or po== BcA AC B. (571) 



" 



It is evident that if p c has a value greater than that determined by 

 this equation, such a film will develop into a larger mass ; if p c has a 

 less value, such a film will tend to diminish. Hence, when 



the phases A and B have a stable surface of contact. 



* If we represent phases by the position of points in such a manner that coexistent 

 phases (in the sense in which the term is used on page 96) are represented by the same 

 point, and allow ourselves, for brevity, to speak of the phases as having the positions of 

 the points by which they are represented, we may say that three coexistent phases are 

 situated where three series of pairs of coexistent phases meet or intersect. If the three 

 phases are all fluid, or when the effects of solidity may be disregarded, two cases are to 

 be distinguished. Either the three series of coexistent phases all intersect, this is 

 when each of the three surface tensions is less than the sum of the two others, or one 

 of the series terminates where the two others intersect, this is where one surface 

 tension is equal to the sum of the others. The series of coexistent phases will be 

 represented by lines or surfaces, according as the phases have one or two independently 

 variable components. Similar relations exist when the number of components is greater, 

 except that they are not capable of geometrical representation without some limitation, 

 as that of constant temperature or pressure or certain constant potentials. 



