J. W. Gihhs — Equilibrium of Heterogeneous Substances. 349 



:^ 2' {a'X^, S.r) , + :^ ^' (a' X^, 6x) 2 = 0, (373 ) 



where tlie suffixed immerals distinouish the expressions relating to 

 the masses on opposite sides of a surface of discontinnity. 



Equation (370) exjiresses tlie mechanical conditions of internal 

 equililirium for a continuous solid under the influence of gravity. If 

 we expand the first term, and set the coefficients of dx, 6y, and 6z 

 separately equal to zero, we obtain 



dX.y^, «Ay, (f-Az, 



~d^ "^ ~dif "^ ~~dl' ~ ' 



dZx, cIZy, dZz, .,, 



~W + "di/^ + Ik^ ~ '' 



The first member of any one of these equations multiplied by dx! dy' 

 dz' evidently represents the sum of the components parallel to one of 

 the axes X, Y, Z of the forces exerted on the six faces of the element 

 dx' dy' dz' by the neighboring elements. 



As the state which we have called the state of reference is arbitrary, 

 it may be convenient for some purposes to make it coincide with the 

 state to which x, y, z relate, and the axes X', Y', Z' with the axes 

 X^ Y.^ Z. The values of Xx,^ . . . Zj, on this particular supposition 

 may be represented by the symbols X^, . . . Zj. Since 



d^v, , ^^ d£y, 



dy dx 



and since, when the states .v, y, z and x', y', z' coincide, and the axes 



X, I", Z, and X, Y\ Z', d^, and d~-, represent displacements 

 which differ only by a rotation, we must have 



Xy = J'x, (375) 



and for similar reasons, 



r, = Zv, Zx = X,. (376) 



The six quantities X^, Yy, Z?, Xy or Y^, Yy or Zy, and Zx ovX^ are 

 called the rectang^dar components of stress, the three first being the 

 longitudinal stresses and the three last the shearing stresses. The 

 mechanical conditions of internal equilibrium for a solid under the 

 influence of gravity may therefore be expressed by the equations 



