BANCROFT. — TERNARY MIXTURES, 337 



represented by the expression V^= F (x,y,z). While the knowledge 

 of the form of this function is necessary to enable us to calculate the 

 volume concentrations of a given solution from our experimental data, 

 it is superfluous in the present discussion. We have (from Formula I.) : 



a log X + /3 log y — (a + /3) log z = log C. 



Now a log F + i8 log F — (a + /3) log V=0, 



alog - + y8 log - - (a + /5) log- = log (7; 



x"-^ 



or 



^a + P 



^ = C, if X, y, and z denote volume concentrations instead of 



having their previous significance. Since a, 3, and C remain unchanged, 

 we find that Equation I represents the series of saturated solutions 

 obtained at constant temperature with any two non-miscible liquids, 

 and a third liquid miscible in all proportions with each of the other 

 two, provided no chemical reaction takes place and provided the react- 

 ing weights of the liquids remain unchanged. It is immaterial whether 

 X, y, and z denote volume concentrations, or concentrations of two of 

 the substances in a constant quantity of the third. 



As has been said, volume concentrations are generally looked upon 

 as the only scientific way of expressing data. This is perfectly nat- 

 ural when we remember that our theoretical ideas have been formed 

 almost entirely upon a study of the gaseous state. It is not a neces- 

 sary method, and in this particular case it is decidedly disadvantageous 

 practically to use volume concentrations. It involves a determination 

 of the density of each solution, increasing the work and bringing in a 

 new source of error. When expressed in volume concentrations all 

 three components vary, and while it is a simple matter to plot three 

 variables in a plane,* I know of no way in which this can be done for 

 the logarithms of these variables. By the method which I have fol- 

 lowed, one constituent can be kept constant, no density determinations 

 are necessary, and there are only two variables. The formula being 

 hyperbolic, by plotting the data on logarithmic co-ordinates one gets a 

 straight line, any variation from which is easily seen, while the con- 

 stants of the curve can be determined from the diagram with more 

 speed and accuracy than by substituting the experimental values in 

 an equation and solving for two unknown quantities. 



The next case to be considered is when we have two partially miscible 



* Gibbs, Tliermodynamisclie Studien, p. 141 ; Eoozeboom, Zeitschr. f. ph. 

 Chem., XII. 369, 1893. 



VOL. XXX. (n. S. XXII.) 22 



