188 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



the fluid (within the envelop), and 2 X denotes a summation with 

 regard to those independently variable components of the fluid of 

 which the solid is composed. Where the solid does not consist of 

 substances which are components, actual or possible (see page 64), 

 of the fluid, this term is of course to be cancelled. 



If we wish to take account of gravity, we may suppose that it acts 

 in the negative direction of the axis of Z. It is evident that the 

 variation of the energy due to gravity for the whole mass considered 

 is simply 



fffgT'te dx'dy'dz', (359) 



where g denotes the force of gravity, and I" the density of the 

 element in the state of reference, and the triple integration, as before, 

 extends throughout the solid. 



We have, then, for the general condition of equilibrium, 



ffft Sr] v ,dx' dy'dz +fffW'x x , S dx'dy'dz' 



f F p SDv+ 2 lt /Vi SDm^ ^ 0. (360) 



The equations of condition to which these variations are subject are : 



(1) that which expresses the constancy of the total entropy, 



fffSthrdafdtfdsf+fifr SN'Ds'+f F SDri = ; (361) 



(2) that which expresses how the value of SDv for any element of 

 the fluid is determined by changes in the solid, 



SDv=-(aSx+/3Sy + -ySz)Ds-v v ,SN'Ds', ' (362) 



where a, /3, y denote the direction cosines of the normal to the 

 surface of the body in the state to which x, y, z relate, Ds the element 

 of the surface in this state corresponding to Ds' in the state of 

 reference, and v v / the volume of an element of the solid divided by 

 its volume in the state of reference ; 



(3) those which express how the values of SDm l} SDm 2 , etc. for 

 any element in the fluid are determined by the changes in the solid, 



SDm 2 = - T^N'Ds', (363) 



etc., 



where I\', IV, etc. denote the separate densities of the several com- 

 ponents in the solid in the state of reference. 



Now, since the variations of entropy are independent of all the 

 other variations, the condition of equilibrium (360), considered with 

 regard to the equation of condition (361), evidently requires that 

 throughout the whole system 



t = const. (364) 



