EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 317 



expressions we may suppose to be derived from the fluid mass. 

 These expressions, therefore, with a change of sign, will represent 

 the increments of entropy and of the quantities of the components 

 in the whole space occupied by the fluid except that which is 

 immediately contiguous to the solid. Since this space may be 

 regarded as constant, the increment of energy in this space may be 

 obtained (according to equation (12)) by multiplying the above 

 expression relating to entropy by t, and those relating to the 

 components by /*/', yw 2 , etc.,* and taking the sum. If to this 

 we add the above expression for the increment of energy near the 

 surface, we obtain the increment of energy for the whole system. 

 Now by (93) we have 



p" = y" + ^y" ~f" Ml Vl ~J~ A t 2 < y2 / ' ~t~ 6 ^ C * 



By this equation and (659), our expression for the total increment of 

 energy in the system may be reduced to the form 



f[e v ' - tnv - A^V/ +p" + (c x + c 2 )<r] SNDa. (660) 



In order that this shall vanish for any values of SN, it is necessary 

 that the coefficient of 8NDs shall vanish. This gives for the con- 

 dition of equilibrium 



^ Yi 



This equation is identical with (387), with the exception of the term 

 containing o-, which vanishes when the surface is plane.t 



We may also observe that when the solid has no stresses except an 

 isotropic pressure, if the quantity represented by a- is equal to the true 

 tension of the surface, p" ' + (c 1 + c^)ar will represent the pressure in 

 the interior of the solid, and the second member of the equation will 

 represent (see equation (93)) the value of the potential in the solid 

 for the substance of which it consists. In this case, therefore, the 

 equation reduces to 



that is, it expresses the equality of the potentials for the substance of 



*The potential fj^" is marked by double accents in order to indicate that its value 

 is to be determined in the fluid mass, and to distinguish it from the potential ft/ 

 relating to the solid mass (when this is in a state of isotropic stress), which, as we 

 shall see, may not always have the same value. The other potentials /-Uj, etc., have 

 the same values as in (659), and consist of two classes, one of which relates to sub- 

 stances which are components of the fluid mass (these might be marked by the double 

 accents), and the other relates to substances found only at the surface of discontinuity. 

 The expressions to be multiplied by the potentials of this latter class all have the 

 value zero. 



fin equation (387), the density of the solid is denoted by F, which is therefore 

 equivalent to ?/ in (661 ). 



