324 J. W. Gihbs on Graphieid Methoch in the 



these four equations will evidently be equivalent to the three inde- 

 {)ondent equations (1), (2) and (3), combined with the assum})tion 

 which we have just made. For a liquid, then, f, instead of being a 

 function of two quantities v and ?/, is a function of r] alone, — t is also 

 a function of // alone, being equal to the differential co-efficient of the 

 function £ ; that is, the value of one of the three quantities i, £ and ?/, 

 is sufficient to determine the other two. The value of w, moreover, is 

 fixed without reference to the values of ^, f and ?/ (so long as these do 

 not pass the limits of values possible for liquidity) ; while 'p does not 

 enter into the equations, i. e., p may have any value (within certain 

 limits) without affecting the values of ^, f, ?/ or v. If the body change 

 its state, continuing always liquid, the value of TFfor such a change 

 is 0, and that of // is determined by the values of any one of the 

 three quantities <, f and //. It is, therefore, the relations between *, f, 

 ?; and i/, for which a graphical expression is to be sought ; a method, 

 therefore, in which the co-ordinates of the diagram are made equal 

 to the volume and pressure, is totally inapplicable to this particu- 

 lar case; y and^^ are indeed the only two of the five functions of the 

 state of the body, ??, jo, #, f and ?/, which have no relations either to 

 each other, or to the other three, or to the quantities W and H, to be 

 expressed.* The values of v and p do not really determine the state 

 of an incompressible fluid, — the values of #, £ and 77 are still left 

 undetei-mined, so that through every point in the volume-pressure 

 diaoram which represents the liquid there must pass (in general) an 

 infinite number of isothermals, isodynamics and isentropics. The 

 character of this part of the diagram is as follows: — the states of 

 licjuidity are represented by the points of a line parallel to the axis of 

 ])ressures, and the isothermals, isodynamics and isentropics, which 

 cross the field of partial vaporization and meet this line, turn upward 

 and follow its course.f 



In the entropy-temperature diagram the relations of t, £ and ;/ are 

 distinctly visible. The line of liquidity is a curve AB (fig. 5) deter- 

 mined by the relation between t and //. This curve is also an iso- 



* That is, V and p have no such relations to the other quantities, as are expressible 

 by equations ; p, however, cannot be less than a certain function of t. 



f All these difficulties are of course removed when the differences of volume of the 

 liquid at different temperatures are rendered appreciable on the volume-pressure 

 diagram. This can be done in various ways, — among others, by choosing as the body 

 to which V, etc., refer, a sufficiently large quantity of the fluid. But, however we do it, 

 we must evidently give up the possibility of representing the body in the state of vapor 

 in the same diagram without making its dimensions enormous. 



