J. W. Gihhs — Eqidlibrium of Heterogeneous Substances. 4U9 



By diliereutiation we obtain 



(r — a;) dx + x dr =i 0, 

 and dv' ■=. n x^ dr -\- {2 tt r x — tt x^) dx ; 



whence {r — x) dv' = — tc r x^ dr. (536) 



By means of this rehxtion, the condition of stability may be reduced 

 to the form 



dp' dp" 2d(J . , „ r — X 

 'ch'~Wr dv' ^ ^^' ~^^ ' 1FV^~^^' ^'^ ^ 



liet us noAV suppose that the temperature and all the potentials ex- 

 cept one, //,, are to be regarded as constant. This will be the case 

 when one of the homogeneous masses is very large and contains all 

 the components of the system except one, or when both these 

 masses are very large and there is a single substance at the surface 

 of discontinuity^ Avhich is not a component of either ; also when 

 the whole system contains but a single component, and is exposed 

 to a constant temperature at its surface. Condition (537) will re- 

 duce by (98) and (508) to the form 



But y^' v' ^ y^' v" -\- I\s 



(the total quantity of the component specified by the suffix) must be 



constant ; therefore, since 



2 

 dv" = — f?y', and ds =: - dv', 



By this equation, the condition of stability is brought to the form 



(-'--'+^^y 



X — r 



When the substance specified by the suffix is a component of either 



of the homogeneous masses, the terms and s -= — * may generally 



be neglected. When it is not a component of either, the terms ;Ki', 



yi", v' -j—^, ""^~ ™^y ^^ course be cancelled, but we must not 



apply the formula to cases in which the substance spreads over the 

 diaphragm separating the homogeneous masses. 



