444 J. W. Gibhs — Equilibrium of Heterogeneous Substances. 



tern.* But before we consider the equations of condition in detail, 

 we may divide the condition of equilibrium (600) into the three condi- 

 tions 



ft dBif +ft dBff = 0, (601) 



- fjy SJ)v + / o- SBs + fg 6z Bm^^fg dz Dm'' = 0, (602) 



y>, 6D7n\ +f/t, 6Dm\ -^fgzdDni^, +fgz6Dm\ 



+ ///. 6Dm\+fiJ^ SDml+fgz6Dml+/gzSJ)m% 



+ 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 sepa- 

 rately. 



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 



fdJJif+fSBrf — 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 tliermal 

 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 



—fp 6Dv +yo- dDs+fgy 6zl)v -\-fg rdzDs=0. (606) 



It will be observed that this condition has the same form as if the 

 difterent fluids were separated by heavy and elastic membranes with- 

 out rigidity and having at every point a tension uniform in all direc- 

 tions in the plane of the surface. The variations in this formula. 



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

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

 continuity was to be neglected, by regarding the sj^tem 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 discontinuitj' between solids and fluids, and we wish to limit ourselves at 

 present to the consideration of the equilibrium of fluid masses, we sliall give up the 

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

 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. 



