SURFACES OF DISCONTINUITY 651 



lines in which the section by the paper cuts the exterior envelop 

 of the whole system by the letters e,f, g.) Then 



So-s = CAD-hc + (TBD-ca -\- (TcD-oh + oTBc-ae + (TcA-hJ + (TAB-cg, 



since the lengths of the curvilinear lines be, ca, ah, ae, bf, eg, 

 are equal to the areas of the respective cylindrical dividing 

 surfaces for that part of the system which lies between two 

 sections unit distance apart. Also 



Xpv = Pa -fbeg + Pb ■ geae + p c • eabf + po • abc. 



Now let us subtract from Zas — Xpv the quantity 



(TBc-de + (TcA-df + (TAB-dg — pA-fdg — PB-gde — pc-edf 



which is unchanged in value by any variation of the surfaces 

 A-D, B-D, C-D. The result of this subtraction is 



(TAD'be + (TsD-ea + crcD-ab — aBc-dd — ccA-bd — CAs-cd 

 — (pD-abe — pA-bcd — pa-ead — pc-ahd). 



This is the quantity Ws — Wv of page 292, and since it differs 

 from Xcrs — 2py by a quantity which is unaltered by any 

 variation of the surfaces A-D, B-D, C-D, it is also a minimum 

 for a stable configuration provided the tensions and pressure 

 are given. This leads directly to Gibbs' equation [629]. In 

 order to grasp what Gibbs is doing in the subsequent portion 

 of page 292, let us consider what would happen to the equilib- 

 rium configuration which involves a mass of the phase D were 

 the six functions (TBc(.t,iJ.), . . . <TAD{t,iJ^) to be changed to slightly 

 different functions of t, m, H2, . . ., say (rBc'{t,n), . . . aAD'(t,fx), 

 while the pressures still retained the same functional forms as 

 before. This would involve a slightly different configuration, 

 causing a change in the areas to Sbc + dsBc, ■ • • Sad + dsAo, 

 and in the volumes to Va + dvA, ... if equilibrium is to be 

 preserved. For this configuration we should have 



. Ws = (Tad'(Sad + dSAo) + . . . —(TBciSBC + dsac) — . . . 



W/ = Pd(vd + dvo) — Pa(va + dvA) — . . ., 



