280 Prof. J. G. MacGregor on the Calculation of 

 which, in the case of mixtures of equal volumes, becomes 



f=MVi), (3) 



^ =fc(V f ). '..... (4) 



I determined ot l and a 2 from these equations by the fol- 

 lowing graphical process: — Equation (3) was employed by 

 drawing, from experimental data, for simple solutions of elec- 

 trolyte 1, a curve with values of the concentration of the ions 

 (a/V) as abscissas and corresponding values of the dilution (V) 

 as ordinates. This curve was drawn once for all and was used 

 in all determinations. The curve embodying equation (4) 

 had to be drawn anew for each mixture examined. If this 

 mixture was formed of solutions containing n Y and n 2 gramme- 

 molecules pei- unit volume of electrolytes 1 and 2 respectively, 

 the curve had as abscissas the concentrations of ions of a series 

 of simple solutions of electrolyte 2, and as ordinates, since 

 Bender's mixtures were mixtures of equal volumes, n 2 /n v 

 times the corresponding values of the dilutions. Equations 

 (1) and (2) were applied by finding, by inspection, two points, 

 one in each curve, having a common abscissa (a!i/V 1 = a 2 /V 2 ), 



and having ordinates (Y"{~and — V 2 respectively) of such 



Hl 

 magnitude as to have a sum equal to p times the sum of the 



ordinates of the points on the curves determined by the dilu- 

 tions (V/ and V 2 respectively) before mixing. The value of 

 the abscissa common to the two points thus determined gives 

 the concentration of ions of both constituents in the mixture. 

 The corresponding ordinate of the first curve, and that of the 

 second curve multiplied by njn^ give the dilutions (Vi and 

 V 2 ) of the constituents in the mixture. The products of the 

 common value of a/V into V 1 and Y 2 are the required values 

 of a 1 and a 2 respectively. 



It will be obvious that the values of a x and a 2 for a solution 

 containing two electrolytes with a common ion may be deter- 

 mined in this way, whether it has been formed by the mixing 

 of two simple solutions or not. It may always be imagined 

 to have been formed in this way ; and if data are not available 

 for the determination of p } special density measurements may 

 be made. 



