J. W. Glhhs — Equilibriiini of Heterogeneous Substances. 363 



known, we may obtain by differentiation the values of ^, JC^,, . . . Zz, 

 in terms of the former quantities, wliich will give eleven indepondcnt 

 relations between the twenty-one quantities 



dx dz 



^^'' ''^'' dx" ' ' ' dz'' ^^ '' ' ' ' ' i'^^^) 



which are all that exist, since ten of these quantities are independent. 

 All these equations may also involve variables which express the 

 composition of the body, when that is capable of continuous varia- 

 tion. 



If we use the symbol if-y, to denote the value of // (as defined on 

 pages 144, 145) for any element of a solid divided by the volume of 

 the element in the state of reference, we shall have 



tl\,= ey,~ f r/y,. (416) 



The equation (356) may l)e reduced to the form 



S,/'y, = - vv, St + ^- ^' (^A\, S~^. (417) 



Therefore, if we know the value of i/\-, in terms of the variables t, 



-yy, , . . . -yii together with those which express the composition of 



the body, we may obtain by differentiation the value« of ?/v», Ax» , 

 . . . Zz, in terms of the same variables. This will make eleven inde- 

 pendent relations between the same quantities as before, except that 

 we shall have y-y, instead of fv»- Oi" if we eliminate ?/-y, by means 

 of equation (416), we shall obtain eleven independent equations be- 

 tween the quantities in (415) and those which express the composi- 

 tion of the body. An equation, therefore, which determines the 



dx dz 



value of ?/-y, , as a function of the quantities ^, --— , , ...---,, and the 



\.tJb (.tZ 



quantities which express the composition of the body when it is capa- 

 ble of continuous variation, is a fundamental equation for the kind of 

 solid to which it relates. 



In the discussion of the conditions of equilibrium of a solid, we 

 might have started with the principle that it is necessary and sufficient 

 for equilibrium that the temperature shall lie uniform throughout the 

 whole mass in question, and that the variation of the force-function 

 {->!') of the same mass shall be null or negative for any variation in 

 the state of the mass not affecting its temperature. We might have 

 assumed that the value of //' for any same element of the solid is a 



