386 '/ W. Gibbs on a Representation by Surfaces 



the center of gravity of masses proportioned to the masses of the 

 parts of the hody in the two ditFerent states and placed at the points 

 of the primitive surface whicli rei)resent these two states (i. e., which 

 represent the volume, entropy, and energy of the body, if its whole 

 mass were supposed successively in the two homogeneous states which 

 occur in its parts). It will therefore be found upon the straight line 

 which unites these two points. As the pressure and temperature are 

 evidently constant for this line, a single i^lane can be tangent to the 

 derived surface throughout this line and at each end of the line tan- 

 gent to the primitive surface.* If we now imagine the temperature 



* Tt is here shown that, if two different states of the substance are such that they 

 can exist permanently in contact with each other, the points representing these states 

 in the tliermodynamic surface have a common tangent plane. We shall see hereafter 

 that the converse of this is true, — that, if two points in the thermodynamic surface 

 have a common tangent plane, the states represented are such as can permanently 

 exist in contact ; and we shall also see what determines the direction of the discon- 

 tinuous cliange which occurs wlien two different states of the same pressure and tem- 

 perature, for which the condition of a common tangent plane is not satisfied, are 

 brought into contact. 



It is easy to express this condition analj^tically. Resolving it into the conditions, 

 that the tangent planes shall be parallel, and that they shall cut the axis of e at the 

 same point, we have the equations 



P'=l>", (a) 



r= t'\ (/?) 



£'— tfri' + p'v'= e''—1"7]"+ p"v", (y) 



where the letters which refer to the different states are distinguished by accents. If 

 there are three states which can exist in contact, we must have for these states, 



e'_ t'ri'+p'v'= e"- t"v" + 'p"v"- t'" - t'"rf" +p"'v"' . 



These results are interesting, as they show us how we might foresee whether two 

 given states of a substance of the same pressure and temperature, can or cannot exist 

 in contact. It is indeed true, that the values of f and -q cannot like those of v^ p, and 

 t be ascertained by mere measurements upon the substance while in the two states in 

 question. It is necessary, in order to find the value of e"— e' or ?/'— if-, to carry out 

 measurements uijon a process by which the substance is brought from one state to the 

 other, hut this need 7iot he hy a process in which the two given states shall be found in con- 

 tact, and in some cases at least it may be done by processes in which the body remains 

 always homogeneous in state. For we know by the experiments of Dr. Andrews 

 (Phil. Trans., vol. 159, p. 575), that carbonic acid may be carried from any of the 

 states which we usually call liquid to any of tliose which we usually call gas, without 

 losing its homogeneity. Now, if we had so carried it from a state of liquidity to a 

 state of gas of the same pressure and temperature, making the proper measurements 

 in the process, we should be able to foretell what would occur if these two states of 

 the substance should be brought together, — whether evaporation would take place, or 

 condensation, or whether they would remain unchanged in contact, — although we had 



