J. W. Gibbs — J^qnlUbrmyii of Heterocieneous l^nhstances. 345 



no beat to be impiivted to tbe element, we sball bave, on multiplying 

 by (/a;' dy dz\ 



dsy, dx' d)j' dz' =. Xx, 6-^, dx' dy' dz'. 



Now tbe first member of tbis equation evidently represents tbe work 

 done upon the element by the surrounding elements ; the second 

 member must therefore have tbe same value. Since we must regard 

 tbe forces acting on opposite faces of tbe elementary parallelopiped as 

 equal and opposite, tbe Avbole work done will be zero except for the 



foce which moves parallel to A". And since d~j—, dx represents the 



distance moved by this face, A'x. dy' dz' must be equal to tbe com- 

 ponent parallel to A' of the force acting upon this face. In general, 

 therefore, if by tbe positive side of a surface for which x' is constant 

 we understand the side on which x has the greater value, we may say 

 that Ax( denotes the component parallel to X of the force exerted by 

 tbe matter on the positive side of a surface for which x' is constant 

 upon the matter on the negative side of that surface per unit of the 

 surface measured in tbe state of reference. The same may be said, 

 mutatis mutandis, of tbe other symbols of tbe same type. 



It will be convenient to use ^ and 2' to denote summation with 

 respect to quantities relating to the axes A, IT, Z, and to tbe axes 

 X', Y', Z', respectively. With tbis understanding we may write 



Se„ = t d,j„ + ^ 2' (a^x, (J^). (356) 



This is the complete value of the variation of e^, for a given element 

 of tbe solid. If we multiply by dx' dy' dz', and take the integral for 

 tbe whole body, we sball obtain the value of tbe variation of the total 

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

 But if we suppose the body to be ino-eased or diminished in substance 

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

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

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

 the integral 



fsy, 6jV' Ds' 



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

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

 the substance added or taken away) measured normally and out- 

 ward in tbe state of reference. The complete value of tbe variation 

 of tbe intrinsic energy of the solid is therefore 



