346 J. W. Gibhs — Eqa'dibrium of Heterogeneous iSuhstances. 



ffftd,j^,,dx'dy'dz' + fff 2:2'L\\,S~) dx'dy'dz' + f Sy, SN'Bs'. (35 7 ) 



This is entirely independent of any supposition in regard to the 

 homogeneity of the solid. 



To obtain the conditions of equilibrium for solid and fluid masses 

 in contact, we should make the variation of the energy of tlie whole 

 equal to or greater than zero. But since we have already examined 

 the conditions of equilibrium for fluids, we need here only seek the 

 conditions of equilibrium for the interior of a solid mass and for the 

 surfaces where it comes in contact with fluids. For this it will be 

 necessary to consider the variations of the energy of the fluids only 

 so far as they are immediately connected with the changes in the 

 solid. We may suppose the solid with so much of the fluid as is in 

 close proximity to it to be enclosed in a fixed envelop, which is 

 impermeable to matter and to heat, and to which the solid is firmly 

 attached wherever they meet. We may also suppose that in the 

 narrow space or sj)aces between the solid and the envelop, which are 

 filled with fluid, there is no motion of matter or transmission of heat 

 across any surfaces which can be generated by moving normals to the 

 surface of the solid, since the terms in the condition of equilibrium 

 relating to such processes may be cancelled on account of the internal 

 equilibrium of the fluids. It will be observed that this method is 

 perfectly applicable to the case in which a fluid mass is entirely 

 enclosed in a solid. A detached portion of the envelop will then be 

 necessary to separate the great mass of the fluid from the small 

 portion adjacent to the solid, which alone we have to consider. Now 

 the variation of the energy of the fluid mass will be, by equation 



(13), 



J'H SDrj - /■•> 6Bv + ^^ , f pi , dJJm^, (358) 



where /^ denotes an integration extending over all the elements of 

 the fluid (within the envelop), and 2^ denotes a summation with 

 regard to those independently variable components of the fluid of 

 which the solid is composed. Where the solid does not consist of 

 substances which are. components, actual or possible (see page 117), 

 of the fluid, this term is of course to be cancelled. 



If we wish to take account of gravity, we may suppose that it acts 

 in the negative direction of the axis of Z. It is evident that the 

 variation of the energy due to gravity for the whole mass considered 

 is simply 



fj'fff ^' ^^^ d^ dy' dz', (359) 



where g denotes the force of gravity, and F' the density of the 



