222 RELATION OF PHYSICAL PROPERTIES OF AQUEOUS SOLUTIONS 
or I should have determined all the values of @ required at 
the outset, and checked them by comparison with one another. 
I have determined the ionization-constants (k% and 7) in all 
cases in which more than two observations of a property on 
solutions of sufficient dilution were available, by the method of 
least squares. The constants thus determined and used in the 
calculations are tabulated below. In all cases the available 
observations had been made on solutions of such great concen- 
tration that the values of the constants obtained cannot be 
regarded as exact; but the calculations may serve as a test of 
the general applicability of the expression referred to above. 
The only available observations, as far as ] know, on solutions 
of sufficient dilution for the determination of the ionization- 
constants and the limits of concentration within which the 
above expression is applicable, are those by Kohlrausch and 
Hallwachs* on the specific gravity of dilute solutions, from 
which two of my students have undertaken to determine the 
density-constants for the salts and acids examined. 
With regard to the observations which I used in determining 
the various ionization-constants, the following statements should 
be made :— 
Bender’s determinations of density (@. e. specific gravity 
referred to water at 4°C.) were made at 15°C., but were 
readily reduced to 18° by the aid of his observations on the 
thermal expansion between 15° and 20° of the same solutions. 
According to his statement, the fourth place of decimals in his 
values may be in error by +2 or +3. The density of water 
was taken to be 0:99863. 
Bender’s determinations of thermal expansion are for the 
interval between 15° and 20° C., and will therefore be sufficiently 
nearly proportional to the coefficients of expansion at 18° for 
my purpose. He considers that they may be in error by +2 in 
the sixth place of decimals. On plotting his observations, how- 
ever, it becomes obvious that they do not all attain this degree” 
* Wied. Ann. lili. (1894) p. 14. 
