J. TT. Gibbs — Eqxdlibriuin of Heterogeneous Substances. 351 



(«')i = - (>^')2, (/^')i = - (A")2, (r')i = - (/)2, 



we may also write 



etc., j 



where tlie signs of a' , (3' , y' may be determined by tlie normal on 

 either side of the surface of discontinuity. 



Equation (371) expresses the mechanical condition of equilibrium 

 for a surface where the solid meets a fluid. It involves the separate 

 equations 



Ds 



a X^, + ft' Xy, + / A-",, = - ap ^^^, 



a' T^, + ff Y„ + r' Vy.. = - Pp ^-, 



Ds 



a' Zx, -f- fi' Zy, + y' Zy,, - - yp -^, 



(381) 



Ds 



the fraction -^^-, denotino; the ratio of the areas of the same element 

 Ds ^ 



of the surface in the strained and unstrained states of the solid. 



These equations evidently express that the force exerted by the 



interior of the solid upon an element of its surface, and determined 



by the strain of the solid, must be normal to the sui-face and equal 



(but acting in the opposite direction) to the pressure exerted by the 



fluid upon the same element of surface. 



If we wish to replace a and Ds by a', ft', y', and the quantities 



which express the strain of the element, we may make use of the 



following considerations. The product a Ds is the projection of the 



Ds 



element Ds on the Y-Z plane. Now since the I'atio — r— , is indepen- 



Ds 



dent of the form of the element, we may suppose that it has any 



convenient form. Let it be bounded by the three surfaces x' zz const., 



y' = const., z' ■=. const., and let the parts of each of these surfaces 



included by the two others with the surface of the body be denoted 



by i, 31, and X, or by L', M' and X' , according as we have reference 



to the strained or unstrained state of the body. The areas of Z', M' , 



and X' are evidently a' Ds' , ft' Ds', and y' Ds' ; and the sum of the 



projections of X, iHiT and TV" upon any plane is equal to the projection 



of Ds upon that plane, since L, M, and X with Ds include a solid 



figure. (In propositions of this kind the sides of surfaces must be 



distinguished. If the normal to Ds falls outward from the small 



solid figure, the normals to L, M, and X must fall inward, and vice 



Trans. Conn. Acad., Vol. III. 45 May, 1877. 



