278 



EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



by the fundamental equation of the mass or surface concerned, or may 

 be immediately derived from it. See pp. 85-89 and 229-231.) 



The variations in (600) are subject to the conditions which arise 

 from the nature of the system and from the supposition that the 

 changes in the system are not such as to affect external bodies. This 

 supposition is necessary, unless we are to consider the variations in 

 the state of the external bodies, and is evidently allowable in seeking 

 the conditions of equilibrium which relate to the interior of the 

 system.* But before we consider the equations of condition in 

 detail, we may divide the condition of equilibrium (600) into the 

 three conditions 



(601) 



(602) 



-fp SD 



v 



fo-SDs + fgSz Dm v +fgSz Dm 8 = 0, 



1+fgztDml 



+ fgz SDrnl + fgz 8 Dm 8 



+ etc. = 0. 



(603) 



For the variations which occur in any one of the three are evidently 

 independent of those which occur in the other two, and the equations 

 of condition will relate to one or another of these conditions 

 separately. 



The variations in condition (601) are subject to the condition that 

 the entropy of the whole system shall remain constant. This may be 

 expressed by the equation 



fSDr} v +fSDr) 8 = 0. (604) 



To satisfy the condition thus limited it is necessary and sufficient that 



t = const. (605) 



throughout the whole system, which is the condition of thermal 

 equilibrium. 



The conditions of mechanical equilibrium, or those that relate to 

 the possible deformation of the system, are contained in (602), which 

 may also be written 



zDs = Q. (606; 



*We have sometimes given a physical expression to a supposition of this kind, 

 problems in which the peculiar condition of matter in the vicinity of surfaces 

 discontinuity was to be neglected, by regarding the system as surrounded by a rigid and 

 impermeable envelop. But the more exact treatment which we are now to give the 

 problem of equilibrium would require us to take account of the influence of the envelop 

 on the immediately adjacent matter. Since this involves the consideration of surfaces 

 of discontinuity between solids and fluids, and we wish to limit ourselves at present 

 to the consideration of the equilibrium of fluid masses, we shall give up the conception 

 of an impermeable envelop, and regard the system as bounded simply by an imaginary 

 surface, which is not a surface of discontinuity. The variations of the system must be 

 such as do not deform the surface, nor affect the matter external to it. 



