EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 187 



the whole body, we shall obtain the value of the variation of the total 

 energy of the body, when this is supposed invariable in substance. 

 But if we suppose the body to be increased or diminished in substance 

 at its surface (the increment being continuous in nature and state 

 with the part of the body to which it is joined), to obtain the com- 

 plete value of the variation of the energy of the body, we must add 

 the integral 



in which Ds' denotes an element of the surface measured in the state 

 of reference, and 8N' the change in position of this surface (due to 

 the substance added or taken away) measured normally and outward 

 in the state of reference. The complete value of the variation of the 

 intrinsic energy of the solid is therefore 



ffft ^'dx'dy'dz' +fff^'x^)dxdy'dz f +f v ,SN'Ds'. (357) 



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 the 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 spaces 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), 



f*t SDn-f*p cSDv+Sj/Vi SDm lt (358) 



where y F denotes an integration extending over all the elements of 



