638 RICE ART. L 



The initial value of this function is the quantity W defined in 

 equation [573]. Its final value is zero. If we differentiate it 

 with respect to x we find that 



df{x) = [(Tag dSi{x) + (Tbc dSiix) — (Tab dsaix) 

 + Pa dvi{x) + Pb dviix) — X dv3{x)] 

 — V3(x)dx, 



and by the fact that there is equilibrium at every stage of this 

 process, which is conceived to take place reversibly, the expres- 

 sion inside the square brackets on the right-hand side is zero. 

 Hence 



df{x) = —Vi{x)dx. 



Integrating we obtain 



f(pc") - Kpc') = - r^" v,(x) dx. 



J PC 



Since the upper limit pc" is larger than pc, as we have men- 

 tioned above, and since V3{x) is a positive quantity throughout, 

 the integral on the right-hand side must be positive also. 

 Therefore the expression on the right-hand side is negative. 

 Hence 



SiPc') >Kpc"). 



But/(pc") is zero, since at the final stage Si (a:), S2 (a;), . . . and 

 V2,{x) are all zero. Hence /(pc')> or W, is positive. Now W is 

 the energy excess in the initial state of the system over the final 

 state. Since it is positive, the initial state of the system has 

 really more energy than the final state, and moreover it is free 

 energy, as the expression [573] shows. Thus the initial state 

 would be unstable and so would not tend to form. 



The treatment of stability given by Gibbs in this subsection 

 and the one preceding must form an important part of any 

 body of principle from which one may hope to obtain in time 

 a satisfying explanation of the colloidal state. Looking back to 



