280 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



in which -* , -j are determined by the function mentioned, and 



&*>!, So) 2 , by the component of the motion of Ds which lies in the 

 plane of the surface. 



With this understanding, which is also to apply to Sp and So- 

 when contained implicitly in any expression, 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 



fpSDv = Sfp Dv-fSp Dv. 



But it is evident that 



SfpDv=fpSNDs, 



where the second integral relates to the surfaces of discontinuity 

 bounding the volume considered, and SN denotes the normal 

 component of the motion of an element of the surface, measured 

 outward. Hence, 



fpSDv=fpSNDs -fSp Dv. 



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

 system is divided, we may write for the whole system 



fpS Dv=f(p'-p")SN Ds-fSp Dv, (609) 



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

 Ds, and SN is measured toward the side specified by double accents. 

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



f<rSDs = 8 fa- Ds - fSa- Ds, 

 and 



where c x and c z denote the principal curvatures of the surface 

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

 which SN is measured), Dl an element of the perimeter of the surface, 

 and ST 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 



fa- SDs =f<r(c l + c 2 ) 8NDa+f2(<r ST) Dl-fS<r Ds, (610) 

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

 which the surfaces of discontinuity 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 



- / (P f -P") SN Ds +fSp Dv +/<r (c x + c 2 ) 8N Ds 



+/2 (o- ST) Dl -fSa- Ds +fgy Sz Dv +fgT Sz Ds = 0. 



