334 >T. W. Gibbs on Graphical Methods in the 



V = IX Vv-{- V Vl-\-{\ - f.1 - v) Vs, 



V = /' '/F+ ^ '/x 4- (1 - /^ - y) Vs, 



where v and ?; are the vohime and entropy of the mixture. The 

 truth of the first equation is evident. The second may be written 



V - Vs = /^ iVv - V,s) + y {Vl - Vs), 

 or multiplying by t', 



^' iV - 'As) =^ Mt' {?/y- }/,^ + J' f {vl - V.v)- 



The first member of this equation denotes the heat necessary to bring 



the body from the state S to the state of the mixture in question 



under the constant tempel■at^^re t', while the terms of the second 



member denote se})arately the heat necessaiy to vaporize the part //, 



and to liquefy the part r of the body. 



The values of v and // are such as would give the center of gravity 



of masses //, v and \—/.i—v j'h^ced at the points V, L and S.* 



Hence the part of the diagram which represents a mixture of vapor, 



liquid and solid, is the triangle VLS. The pressure and temperature 



are constant for this triangle, i. e., an isopiestic and also an isothermal 



here expand to cover a space. The isodynamics are straight and 



de 

 equidistant for equal differences of energy. For — - =:r — 2^, and 



de 



-zr- =■ t\ both of wdiich are constant throughout the triano;le. 



dv B a 



This case can be but very imperfectly represented in the volume- 

 pressure, or in the entropy-temperature diagram. For all points in 

 the same vertical line in the triangle VLS will, in the volume- 

 pressure diagram, l)e represented by a single point, as having the 

 same volume and pressure. And all the points in the same horizontal 

 line will be represented in the entropy-temperature diagram by a 

 single point, as having the same entropy and temperature. In either 

 diagram, the whole triangle reduces to a straight line. It must 

 reduce to a line in any diagram whatever of constant scale, as its area 

 must become in such a diagram. This must be regarded as a defect 

 in these diagrams, as essentially different states are represented by 

 the same point. In consequence, any circuit within the triangle 



* These points will not be in the same straight line unless 



t' iVv - Vs) ■ t' (vl - V.s) :: Vy — v,s : Vj, - Vg, 

 a condition very unlikely to be fulfilled by any substance. The first and second terms 

 of this proportion denote the heat of vaporization (from the solid state) and that of 

 liquefaction. 



