J. TF[ Gihbs — EqiiiUbriuin of Heterogeneous Sxihstauces. 445 



l)esicle their necessary geomelrical relations, are subject to the condi- 

 tions that the external surface of tlie system, and the lines in which 

 the surfaces of discontinuity meet it, are fixed. The formula may be 

 reduced by any of the usual methods, so as to give the particular 

 conditions of mechanical equilibrium. Perhaps the following method 

 will lead as directly as any to the desired result. 



It will be observetl the quantities affected by d in (606) relate 

 exclusively to the position and size of the elements of volume and 

 surface into which the system is divided, and that the variations 6p 

 and 6a do not enter into the formula either explicitly or implicitly. 

 The equations of condition which concern this formula also relate 

 exclusively to the variations of the system of geometrical elements, 

 and do not contain either 6p or 6<j. Hence, in determining whether 

 the first member of the formula has the value zero for every possible 

 variation of the system of geometrical elements, we may assign to 

 8p and da any values whatever, which may simplify the solution of 

 the problem, without inquiring whether such values are physically 

 possible. 



Now when the system is in its initial state, the pressure jo, in each 

 of the parts into which the system is divided by the surfaces of ten- 

 sion, is a function of the co-ordinates which determine the position of 

 the element Z>w, to which the pressure relates. In the varied state 

 of the system, the element Dv will in general have a different position. 

 Let the variation 6p be determined solely by the change in position 

 of the element Dv. This may be expressed by the equation 



in wliich ^ , i? , ^ are determined by the function mentioned 

 dx ay dz ' 



and dec, 6y^ 6z by the variation of the position of the element Dv. 



Again, in the initial state of the system the tension a, in each of 

 the different sui-faces of discontinuity, is a function of two co-ordinates 

 &?j, Gjg, which determijie the position of the element Ds. In the 

 varied state of the system, this element will in general have a differ- 

 ent position. The change of position may be resolved into a com- 

 ponent lying in the surface and another normal to it. Let the varia- 

 tion da be determined solely by the first of these components of the 

 motion of Ds. This may be expressed by the equation 



f^ da ^ . da ^ 



''''=. to; '""■ + .to/'"- (8«8) 



