426 J. ^fV. Gihhs — Eqnilibrium of Heterogeneous Substances. 



since Sac > «ab, and Sbc > «ab. 



But at the limit, when 



Cac + tfBc = Cab, 

 we see by (561) that 



Sac ^^^ ^ABj anci Sbc ^^ ^ab, 



and therefore W ■=:. 0. 



It should however be observed that in the immediate vicinity of the 

 circle in which the three surfaces of discontinuity intersect, the 

 physical state of each of these surfaces must be affected by the 

 vicinity of the others. We cannot, therefore, rely upon the formula 

 (570) except when the dimensions of the lentiform mass are of sensi- 

 ble magnitude. 



We may conclude that after we pass the limit at which p^ becomes 

 greater thanjo^ and jt?B (supposed equal) lentiform masses of phase C 

 will not be formed until either o'ab^o'ac+c'bc? or p^, — -jo^ becomes so 

 great that the lentiform mass which would be in equilibrium is one 

 of insensible magnitude. [The diminution of the radii with increas- 

 ing values of Pc—l^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 

 disregarded when the lentiform mass is of 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.* 



* 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 152) 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 coex- 

 istent phases are situated where three series of pairs of coexistent phases meet or 

 intersect. If the three phases are all fluid, or wlien the effects of solidity may be 

 disreo'arded. 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 inter- 

 sect 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. SimUar 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. 



