./. W. Glbbs — Equilibrium, of Heterogeneous Substances. 205 



-fp 6Bv -\-fg 6h Urn ^ 0, (220) 



if the bounding surface is unvaried ; 



y7<i 61>m^ + fgh 6Bm^ ^0, if fSDm^ = ; 



(227) 

 y>„ SJ}ni„ + fg h 61>m„ ^ 0, if f 6Dm^ = 0. 



From (225) we may derive the condition of thermal equilibrium, 



«z= Const. (328) 



Condition (226) is evidently the ordinary mechanical condition of 

 equilibrium, and may be transformed by any of the usual methods. 

 We may, for example, apply the formula to such motions as might 

 take place longitudinally within an infinitely narrow tube, terminated 

 at both ends by the external surface of the mass, but otherwise 

 of indeterminate form. If we denote by m the mass, and by v the 

 volume, included in the part of the tube between one end and a 

 transverse section of variable position, the condition will take the 

 form 



— fp ddv + fg Sh dm ^ 0, (229) 



in which the integrations include the whole contents of the tube. 

 Since no motion is possible at the ends of the tube, 



fp Sdv + fdv dp =fd{p Sv) z= 0, (230) 



Again, if we denote by y the density of the fluid, 



dh 

 fg dh dm :=fg -^ Sv y dv =.fg y Sv dh. (231 ) 



By these equations condition (229) may be reduced to the form 



fSv {dp -{- g y dh) ^ 0. (232) 



Therefore, since Sv is arbitrary in value, 



dp = — g y dh, (233) 



which will hold true at any point in the tube, the difierentials being 

 taken with respect to the direction of the tube at that point. There- 

 fore, as the form of the tube is indeterminate, this equation must 

 hold true, without restriction, throughout the whole mass. It evi- 

 dently requires that the pressure shall be a function of the height 

 alone, and that the density shall be equal to the first derivative of 

 this function, divided by — g. 



Conditions (227) contain all that is characteristic of chemical 

 equilibrium. To satisfy these conditions it is necessary and sufficient 



that 



yu J -f- ^ A = Const. \ 



(2-34) 



//„ -f gh =. Const. ) 



