192 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



discontinuity by the matter on one side and determined by its state 

 of strain shall be equal and opposite to that exerted by the matter 

 on the other side. Since 



we may also write 



a (^x')i + P(*T\ + y'(X*\ = a'(*x<) 2 + P(Xv\ + v'(*z') 2 >\ (380 ) 

 etc., J 



where the signs of a', /$', y may be determined by the 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 (381) 



Ds 

 the fraction -=p denoting the ratio of the areas of the same element 



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 surface 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', P, y', and the quantities 

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

 following considerations. The product aDs is the projection of the 



Ds 



element Ds on the Y-Z plane. Now since the ratio jr- f is independent 



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

 form. Let it be bounded by the three surfaces x' = 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 L, M, and N, or 

 by L', M', and N', according as we have reference to the strained or 

 unstrained state of the body. The areas of L', M', and N' are evidently 

 a'Ds', B'Ds', and y'Ds' ; and the sum of the projections of Z, M t and 

 N upon any plane is equal to the projection of Ds upon that plane, 

 since L, M, and N 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 N must fall inward, and vice versa.) Now L' is a right-angled 

 triangle of which the perpendicular sides may be called dy' and dz f . 



