446 J. W. Gihhs — Equil'ibrmyn of Heterogeneous Substances. 



in which , are deteriiiined by the function mentioned, and 



dco , doi)^ 



^ooi, (^Ce?2, by the component of the motion of JJs which lies in the 



plane of the surface. 



With this understanding, which is also to apply to dp and dff 

 when contained implicitly in any expi-ession, we shall proceed to the 

 reduction of the condition (606). 



With respect to any one of the volumes into which the system is 

 divided by the surfaces of discontinuity, we may write 



/p SDv =z Sj'p Dv — f Sp Dv. 

 But it is evident that 



dfpDv=fpdNDs, 

 where the second integral relates to the surfaces of discontinuity 

 bounding the volume considered, and 8N denotes the normal com- 

 ponent of the motion of an element of the surface, measured outward. 

 Hence, 



fp SDv = fp SNDs^fS'pDv. 



Since this equation is true of each separate volume into which the 

 system is divided, we may Avrite for the whole system 



fp 6Dv =f{p'—p") 8NDs - fdp Dv, (609) 



where jo' and p" denote the pressiires on opposite sides of the element 

 Ds, and (JiV^is measured toward the side specified by double accents. 

 Again, for each of the surfaces of discontinuity, taken separately, 



/ 6Ds — dfoBs —fda Ds, 

 and 



SfaDs —fa (c, + cg) 8NI)s-^fa dTDl, 



where Cj and c^ denote the principal curvatures of the surface, 

 (]iositive, when the centers are on the side opposite to that toward 

 which (JiVis measured,) Dl^n element of the perimeter of the surface, 

 and (JZ'the component of the motion of this element which lies in the 

 plane of the surface and is perpendicular to the perimeter, (positive, 

 when it extends the surface). Hence we have for the whole system 



fff6J)sz=fff{c^ 4-C2) 6NDs-\-f2{ffST)Dl-fS0l)s, (610) 

 wdiere the integration of the elements Dl extends to all the lines in 

 which the surfaces of discontiiuiity meet, and the symbol 2 denotes 

 a summation with respect to the several surfaces which meet in such 

 a line. 



By equations (609) and (610), the general condition of mechanical 

 equilibrium is reduced to the form 



