J. TF. Gibhs — Equilihrmm of Heterogeiieons Stihutances. 403 



relate to the first and second states oi' the film, the letters without 

 accents denoting those quantities which have the sanu; value in both 

 states. The difl:erential of this expression when the quantities distin 

 guished by double accents are alone considered variable, and the area 

 of the surface is constant, will reduce by (501) to the form 



(//;- ;/;) dn^' + (/.;; - ^x',) drnf + etc. 



To make this incapable of a negative value, we must have 



/.(" = //;' , unless r/if = 0. 



In virtue of these relations and by equation (502), the expression 

 (519), i. e., (516), will reduce to 



ff" s — ff' s, 



which will be positive or negative according as 



ff" — ff' (520) 



is positive or negative. 



That is, if the tension of the film is less than that of any other film 

 which can exist between the same homogeneous masses (which has 

 therefore the same values of t, /<„ , /u,, , etc.), and which moreover has 

 the same values of the potentials //^ , //;, , etc., so far as it contains the 

 substances to which these relate, then the first film will be stable. 

 But the film will be practically unstable, if any other such film has a 

 less tension. [Compare the expression (141), by which the practical 

 stability of homogeneous masses is tested.] 



It is, however, evidently necessary for the stability of the surface 

 of discontinuity with respect to deformation, that the value of the 

 superficial tension should be positive. Moreover, since we have by 

 (502) for the surface of discontinuity 



and by (93) for the two homogeneoiis masses 



e' — t J/' + p v' ~ /Ja w*„' — /./,, m/ — etc. =: 0, 

 e" - t if + p v" -//„ mj' — yi/j m,," — etc. = 0, 

 if we denote by 



f, 7/, V, m„, wij, etc., tiig, ???;,, etc., 



the total energy, etc. of a composite mass consisting of two such 

 homogeneous masses divided by such a surface of discontinuity, we 

 shall have by addition of these equations 



