GIBBS' PAPERS I AND II 45 



The slopes of the plane are t in the erj plane or planes parallel 

 thereto and —pin the ev plane or any parallel plane. Further 



These two quantities are in general different but at the limit of 

 stability they are equal and in particular at the critical point. 

 Both these quantities are easy to measure. If we have coexist- 

 ent phases the tangent plane is rolling on the surface with con- 

 tact at two points and the successive positions intersect in the 

 line giving the two points of contact and representing the diifer- 

 ent states in which the two phases can exist in different propor- 

 tions at the same pressure and temperature. At the critical 

 point according to van der Waals' equation. 



R R ^ tc Sb 



= = — and — = — • 



V - b 2b pc R 



Hence 



\dt/^ 



t d log V , , 



- = -—^ = 4. (20) 



p d log I 



Now we may experimentally determine the values of p and t for 

 states of coexisting phases and make a graph in which we plot 

 log p against log t. If then van der Waals' equation were satis- 

 fied we should find that as we approached the critical point the 

 slope of the curve approached 4. This value does not, as a 

 matter of fact agree with that found by experiment, which points 

 rather to 5 or 6 or 7. Various modifications of the equation 

 have been proposed by Clausius and others. We could treat 

 any of these proposals by similar methods. No entirely satis- 

 factory equation of state has been proposed. The usefulness of 



