J. W. Gihbs — Eqnilihrinm of Heterogeneous Substances. 887 



cicntly small to be considered uniform throughout in its curvatures 

 and in respect to the state of the surrounding matter, the value of the 

 above expression will be determined bj' the variation of its area 6s 

 and the variations of its principal curvatures 6c ^ and (Jc^, and we 

 may write 



tff* =3 t 6}f -{- yu, 6)/i\ -\- /-/g 6m\ -\- etc. 



+ (?<5.^^+ C\ 6c, + C^6c^, (493) 



or 



6£^ = t 6if + /<i 6w^^ -\- /Yg 6in\ + etc. 



^a6s^\{G,-^C^) 6{c., + e^) + i(6', - C^) 6{c^ - c^\{\%\) 



o", C,, and Cg denoting quantities which are determined by the 

 initial state of the system and position and form of s. The above is 

 the complete value of the variation of t^ for reversible variations of 

 the system. But it is always possible to give such a position to the 

 surface s that C^ + (J.^ shall vanish. 



To show this, it will be convenient to write the equation in the 

 longer form [see (490), (491)] 



6^ ^ t 6}f — /<j 6m , — /<2 6)n„ — etc. 

 ~ 6f:"' + t 6)/" + /<, 6m,'" + /<2 6m,'" + etc. 

 _ Se"" + t 6if"' + /Yj 6m,"" + //a 6'm.^ "" + etc. 

 = o-6s + h{C, + 0,) 6{c, + cs) + HC'i - <^',) %. -e,), (495) 

 i. e., by (482)-(484) and (12), 



6e — t 6?/ — /u, 6m, — ^2 ^^^2 — ^^^' ^ P' ^""' ^ P" ^^"" 

 = 0- (Js + i (C, + Co) 6{c, + C2) + h {C\ - C,) 6{c, - c,). (496) 



From this equation it appears in the first place that the pressure is 

 the same in the two homogeneous masses separated by a plane sur- 

 face of discontinuity. For let us imagine the material system to 

 remain unchanged, while the plane surface s without change of area 

 or of form moves in the direction of its normal. As this does not 

 affect the boundaries of the mass M, 



6s — t 6y — fA, 6m, — yWg 6m2 — etc. = 0. 

 Also 6s = 0, 6{c, + C2) = 0, 6{c, — C2) = 0, and 6?^' = - 6v"". 

 Hence ^:>' =2^'\ when the surface of discontinuity is plane. 



Let us now examine the effect of different positions of the surface *^ 

 in the same material system upon the value of C\ -\- t'g, supposing at 

 first that in the initial state of the system the surface of discontinuity 

 is ]»]nne. Let ns give tlie surface s some particular position. Li the 



