EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 271 



We must notice the difference between this formula and (587). In 

 (593) the quantity of heat Q is determined by the condition that the 

 temperature and pressures shall remain constant. In (587) these 

 conditions are equivalent and insufficient to determine the quantity 

 of heat. The additional condition by which Q is determined may be 

 most simply expressed by saying that the total volume must remain 

 constant. Again, the differential coefficient in (593) is defined by 

 considering p as constant ; in the differential coefficient in (587) p 

 cannot be considered as constant, and no condition is necessary 

 to give the expression a definite value. Yet, notwithstanding the 

 difference of the two cases, it is quite possible to give a single 

 demonstration which shall be applicable to both. This may be done 

 by considering a cycle of operations after the method employed by 

 Sir William Thomson, who first pointed out these relations.* 



The diminution of volume (per unit of surface formed) will be 



(594) 



y y \p/ t 



and the work done (per unit of surface formed) by the external 

 bodies which maintain the pressure constant will be 



da\ ( da- \ 



j-) = -(;JT- -) (595) 



dp/ t \dlogp/t 



Compare equation (592). 



The values of Q and W may also be expressed in terms of quan- 

 tities relating to the ordinary components. By substitution in (593) 

 and (595) of the values of the differential coefficients which are given 

 by (581), we obtain 



<2=-*f, w *i* ( 596 > 



where A, B, and C represent the expressions indicated by (582)-(584). 

 It will be observed that the values of Q and W are in general infinite 

 for the surface of discontinuity between coexistent phases which 

 differ infinitesimally in composition, and change sign with the quantity 

 A. When the phases are absolutely identical in composition, it is not 

 in general possible to counteract the effect of extension of the surface 

 of discontinuity by any supply of heat. For the matter at the surface 

 will not in general have the same composition as the homogeneous 

 masses, and the matter required for the increased surface cannot be 

 obtained from these masses without altering their phase. The infinite 

 values of Q and W are explained by the fact that when the phases 

 are nearly identical in composition, the extension of the surface of 



*See Proc. Hoy. Soc., vol. ix, p. 255 (June, 1858); or Phil. Mag., ser. 4, vol. xvii, 

 p. 61. 



