SOLUTIONS.—MACGREGOR. 69 
The first equation is given by the law of kinetic equili- 
brium. It may be expressed as follows: The regional ionie 
concentrations of the two electrolytes, 7. e, the numbers of their 
free gramme-ions per unit volume of their respective regions, are 
equal. The second states that the volume of the solution is 
equal to the sum of the volumes of the regions of the respective 
electrolytes. The third and fourth assert that the regional 
ionic concentrations are functions of the respective regional 
dilutions. 
As f, and f, are very complex functions, these equations 
could not be solved algebraically even if the functions were 
known. They can be solved graphically, however, without 
actually determining what the functions are. 
For this purpose we first find, from conductivity obser- 
vations made on simple solutions of 1 and 2 respectively, 
corresponding values of dilution and ionic concentration for a 
sufficient number of solutions of each, and plot dilution-ionic- 
concentration curves, 2. é, curves with dilutions as ordinates 
and ionic-concentrations as abscissee. To get precise values of 
the ionization coefficients for the complex solutions, these curves 
must be accurately drawn. They have, very roughly speaking, 
the shape of rectangular hyperbolas, and thus, both at great 
dilution and at great concentration, have but slight curvature, 
while at moderate dilution they have very rapid curvature. In 
working with solutions at moderate dilution therefore, it is 
necessary to have a considerable number of corresponding 
values of dilution and ionie concentration, in order to plot the 
curves accurately. When but few are available, it is helpful to 
plot first a concentration-ionic-concentration curve, 7. é@, one 
having concentrations of solutions as ordinates and ionic-con- 
centrations as abscisse. As the dilution-ionic-concentration 
curves are something like rectangular hyperbolas, the concen- 
tration-ionic-cuncentration curves have comparatively slight 
curvature,and thus lend themselves readily to interpolation. 
Corresponding values of concentration and ionic concentration 
obtained from these curves, when the concentrations are trans- 
formed into dilutions, may be used to eke out the values 
