46 IONIZATION COEFFICIENTS OF CERTAIN 
The above table shews the degree of accuracy with which 
the depression of the freezing-point can be calculated in the case 
of simple solutions. If the ionization coefficients for the mix- 
tures are determined by Prof. MacGregor’s method as closely as 
they are for the simple solutions by putting a=,/u., the differ- 
ences between the calculated and observed values of the depres- 
sions in the case of the mixtures may be expected to be no 
greater than those of the above table. 
Depression of the Freezing-point by the Mixtures. 
The following Table contains the observed and calculated 
values of the depression of the freezing-point in the case of the 
mixtures examined. The observations were made in the manner 
described above. The calculations were made by the following 
formula for which I am indebted to Professor MacGregor :— 
A=1.86 (l+a,+a,) N/2 
where a, and a, are the ionization coefticients of the respective 
electrolytes in the mixture and N the number of gramme-equiva- 
valents per litre in the solutions mixed, which were in all cases 
equimolecular, This expression may be readily obtained as 
follows :—In each litre of the mixture there will be N/4 gramme- 
molecules of each electrolyte. There will therefore be (l—«,) N/4 
and (1 — «,) N/4 undissociated gramme-molecules of the respec- 
tive electrolytes, and, if we assume the ionization in each case 
to be complete, 34, N/4 and 3 «, N/4 free gramme-ions. Hence 
the total number of undissociated gramme-molecules and free 
gramme-ions will be (l+@,+a@,) N/2, and the expression for 
the depression will consequently be as above. 
The first column of Table V gives the concentration of the 
solutions mixed, in gramme-equivalents of anhydrous salt per 
litre at 18°C. The fifth and sixth columns give the ionization 
coefficients of the respective electrolytes in the mixture at 0°C 
as determined by Prof. MacGregor’s method. The second, third 
and fourth give the quantities obtained directly by this method, 
viz., the common concentration of ions, and the dilutions of the 
respective electrolytes in the mixture. (By the concentration of 
