422 J. W. Gibhs — Eqidlihriuni of Heterogeneous Stihstances. 



Moreover, the second of these quantities, if we neglect its sign, is 

 always equal to two-tliirds of the first, as appears from equation (550) 

 and the geometrical relation v'rr^^s. We may therefore write 



W= ias=z^{p' - p") v'. (560) 



On the .Possible Formation at the Surface where two different Homo- 

 geneous Fluids meet of a Fluid of different Phase from either. 



Let A, B, and C be three different fluid phases of matter, which 

 satisfy all the conditions necessary for equilibrium Avhen they meet 

 at plane surfaces. The components of A and B may be the same or 

 diflerent, but C must have no components except such as belong to A 

 or B. Let us suppose masses of the phases A and B to be separated 

 by a very thin sheet of the phase C. This sheet will not necessarily 

 be plane, but the sum of its principal curvatures must be zero. We 

 may treat such a system as consisting simply of masses of the phases 

 A and B with a certain surface of discontinuity, for in our previous 

 discussion there has been nothing to limit the thickness or the nature 

 of the film separating homogeneous masses, except that its thickness 

 has generally been supj^osed to be small in comparison with its radii 

 of curvature. The value of the superficial tension for such a film 

 will be <3'ac+'5'bc5 if we denote by these symbols the tensions of the 

 surfaces of contact of the phases A and C, and B and C, respectively. 

 This not only appears from evident mechanical considerations, but 

 may also be easily verified by equations (502) and (93), the first of 

 which may be regarded as defining the quantity a. This value will 

 not be affected by diminishing the thickness of the film, until the 



envelop to yield to the internal pressure until its contents are increased by v' without 

 materially afEecting its superficial area. If this be done sufficiently slowly, the phase 

 of the mass within will remain constant. (See page 139.) A homogeneous mass of 

 the volume v' and of the desired phase has thus been produced, and tlie work gained 

 is evidently {p'—p")v'. 



Let us suppose that a small aperture is now opened and closed in the envelop so as 

 to let out exactly the volume v' of the mass within, the envelop being pressed inwards 

 in another place so as to diminish its contents by this amount. During the extrusion 

 of the drop and until the orifice is entirely closed, the surface of the drop must adhere 

 to the edge of the orifice, but not elsewhere to the outside surface of the envelop. 

 The work done in forming the surface of the drop will evidently be as or 2{p'—p"}v'. 

 Of this work, the amount {p'—p")v' will be expended in pressing the envelop inward, 

 and the rest in opening and closing the orifice. Both the opening and the closing 

 will be resisted by the capillary tension. If the orifice is circular, it must have, when 

 widest open, the radius determined liy equation (550). 



