-/ W. Gihbs — Equilihriimi of Heterogeneous Substances. 425 



which signifies that the components parallel to EF of the tension 

 cTac and Cbc are together equal to Cab- If we denote by W the 

 amount of work which must be expended in order to form such a 

 lentiform mass as we are considering between masses of indefinite 

 extent having the phases A and B, we may write 



W= M - jV, (562) 



where M denotes the work expended in replacing the surface be- 

 tween A and B by the surfaces between A and C and B and C, and 

 iV denotes the work gained in replacing the masses of phases A and 

 B by the mass of phase C. Then 



31 = (Tac Sac + O'bc Sbc - (5'ab «ab, (563) 



where Sac, He ^ab denote the areas of the three surfaces concerned ; 

 and 



N= V {pc. - pd + V" (pc -Pb), (564) 



where F' and V denote the volumes of the masses of the phases 

 A and B which are replaced. Now by (500), 



Pc—Pa=——, and pc-p^^——' (565) 



(566) 



We have also the geometrical relations 



F' = I ;r r'2 x' - ^ n K'' (r' - x'), \ 

 V" = ^7t r" ^x" - iTtR^ (r" - x"). \ 



By substitution we obtain 



/*' — x' 



N= f 7t o\c r' x' — ^7t R~ o\c ,-^ 



r 



r" x" 



+ f TT o'bc r" x" ^In R^ o'bc r, — , (567) 



and by (561), 



N= \7t Gt^^r' x' ^%n (Tbc r" x" - ^ tt R^ (Tab- ^ (568) 



Since 



2 n r' x •=. Sac) '^ ^ ^" ^" = He-, ^ ^^ = ^ab, 



we may write 



N'= f ((Tac .''ac + O'bc «bc - ^-ab «ab)- (569) 



(The reader will observe that the ratio of M and iV is the same as 

 that of the corresponding quantities in the case of the spherical mass 

 treated on pages 416-422,) We have therefore 



"R^= i (^Ac Sac + O'bc ^bc - ^-ab Sab)- (570) 



This value is positive so long as 



