J. W. Gibbs — Equilihrium of Heterogeneous Substances. 459 



Let us denote by y^, ^b? ^c the portions of v-o which were originally 

 occupied by the masses A, B, C, respectively, by s^a, s^b, Sdc? the 

 areas of the surfaces specified per unit of length of the mass D, and 

 hy ^s^B, Hci ^cKi the areas of the surfaces specified which were replaced 

 by the mass D per unit of its length. In numerical value, t^^, v^, Vq 

 will be equal to the areas of the curvilinear triangles bed, cad, 

 ab d; and Sp^, s^b? ^dc? ^ab? *bci -^ca to the lengths of the lines he, c a, 

 ab, c d, ad, b d. Also let 



f' S — ^ ^DA ^VA "T ^DB ^DB T" '^'dC ^DC — (^ AB ■^AB ^]iC ^BC — ^CA *'cA5 (626) 



and T^ .r=^D Vd — i»A ^a - 2^b Vb — Pc Vc- (627) 



The general condition of mechanical equilibrium for a system of 

 homogeneous masses not influenced by gravity, when the exterior of 

 the whole system is fixed, may be written 



2 {a Ss) - 2{p dv) =: 0. (628) 



[See (606).] If we apjjly this both to the original system consisting 

 of the masses A, B, and C, and to the system modified by the intro- 

 duction of the mass D, and take the difference of the results, suppos- 

 ing the deformation of the system to be the same in each case, we 

 shall have 



O'da OSda ~I~ CTdb O^db "1" ^DC OSpc — (Tab OS^b ~~ <5'bc ^5bc 



~ O-ca ^Sca - Pt> ^Vjy + pi, (Jva + Pb ^V^ -f- Pc 6vc = 0. (629) 



In view of this relation, if we differentiate (626) and (627) regarding 

 all quantities except the pressures as variable, we obtain 



dWs— dWy = SdA C^O'dA + «DB dffoB + «DC cK^DC 



- Sab dffj^s — Sbc ^Cbc — «ca ^(?ca- (630) 



Let us now suppose the system to vary in size, remaining always 

 similar to itself in form, and that the tensions diminish in the same 

 ratio as lines, while the pressures remain constant. Such changes 

 will evidently not impair the equilibrium. Since all the quantities 

 Sda, ^da, Sdb, Cdb, etc. vary in the same ratio, 



SDAf?0-uA=it?((>'DA W^ SDB(^ffT,B=id{(}j)TiSTjB), CtC. (631) 



We have therefore by integration of (630) 



TFs- Trv = i((rDA«DA+^DBSDB + 0'DC«DC — 0'aB«AB-0'bC«BC-0'ca«Ca), (632) 



whence, by (626), 



TFs = 2 Wy, (633) 



The condition of stability for the system when the pressures and 



tensions are regarded as constant, and the position of the surfaces 



