ABSTRACT BY THE AUTHOR. 357 



fluids will depend upon the rigidity of the diaphragm. Even when 

 the diaphragm is permeable to all the components without restriction, 

 equality of pressure in the two fluids is not always necessary for 

 equilibrium. 



Effect of gravity. In a system subject to the action of gravity, 

 the potential for each substance, instead of having a uniform value 

 throughout the system, so far as the substance actually occurs as an 

 independently variable component, will decrease uniformly with 

 increasing height, the difference of its values at different levels being 

 equal to the difference of level multiplied by the force of gravity. 



Fundamental equations. Let e, jy, v, t and p denote respectively 

 the energy, entropy, volume, (absolute) temperature, and pressure of 

 a homogeneous mass, which may be either fluid or solid, provided 

 that it is subject only to hydrostatic pressures, and let m 1} ra 2 , ... ra n 

 denote the quantities of its independently variable components, and 

 fji lt // 2 , ... fJL n the potentials for these components. It is easily shown 

 that e is a function of ij, v, m 1 , ra 2 , ... ra n , and that the complete .value 

 of de is given by the equation 



de = tdq p dv + fi l dm 1 + /z 2 dm 2 . . . -f n n d / m n . (5) 



Now if is known in terms of ;;, v, m 1} ... m n , we can obtain by 

 differentiation t, p, /z x , ... fj. n in terms of the same variables. This 

 will make n + 3 independent known relations between the 2n + 5 

 variables, e, r\, v, ra^ m 2 , ... m n , t, p, JUL V /* 2 , ... /x n . These are all that 

 exist, for of these variables, 7i+2 are evidently independent. Now 

 upon these relations depend a very large class of the properties of 

 the compound considered, we may say in general, all its thermal, 

 mechanical, and chemical properties, so far as active tendencies are 

 concerned, in cases in which the form of the mass does not require 

 consideration. A single equation from which all these relations may 

 be deduced may be called a fundamental equation. An equation 

 between e, 77, v, m l , m 2 , ... m n is a fundamental equation. But there 

 are other equations which possess the same property. 



If we suppose the quantity \fs to be determined for such a mass 

 as we are considering by equation (3), we may obtain by differentiation 

 and comparison with (5) 



d\fs = rjdt p dv + fjL 1 dm l + fJL z dm 2 . . . + fjL n dm n . (6) 



If, then, \fs is known as a function of t, v, m x , ra 2 , ... m n , we can find 

 n> P> Pi* A t 2>-"/ u n in terms of the same variables. If we then 

 substitute for \[s in our original equation its value taken from 

 equation (3) we shall have again n+3 independent relations between 

 the same 2n+5 variables as before. 

 Let 



(7) 



