In order to study quantitatively the acidity of very weak acids and at the same time the effect of a polar aprotic solvent on the basicity of hydroxide ion a Hammett H_ acidity function based on the ionization of 24 substituted anilines and diphenylamines has been established in the system dimethylsulfoxide-water-tetramethylammonium hydroxide. The basicity of hydroxide ion is increased dramatically as the solvent is changed from water to dimethylsulfoxide. The H_ of a 0.011 molar solution of tetramethylammonium hydroxide ranges from 12 in water to 26 in 99.6 mole % dimethylsul-foxide-water, an increase in basicity of fourteen powers of ten. The increase in basicity is due to the increased activity of the hydroxide ion brought about by the reduction in its solvation in the poor anion-solvating solvent, dimethyl sulfoxide and indicates the extensive solvation enjoyed by the hydroxide ion in water. The pKHA values of the indicator acids vary from 13.84 for 2,4-dinitrodiphenylamine to 25.63 for 3-chloroaniline. From a plot of log KHA versus Hammett substituent constants ( ϭ) for six monosubstituted diphenylamines a rho (ρ) value of 4.07 is found. The magnitude of substituent effects on the acidity of aniline appears to be quite similar. The acidities of all the substituted diphenylamines do not follow the above mentioned correlation with Hammett ϭconstants; the pKHA values of 4-amino-, 4-methoxy-, 4-methylsulfonyl-, and 4-nitrodiphenylamine are all less than expected from the Hammett ϭ constants for the substituents in these compounds. The 4-nitro substituent exerts a particularly large acid-strengthening effect on the acidities of aniline and diphenyl amine; the decreases in pKHA being approximately 8.4 and 6.8 pK units, respectively. The pKHA values of 17 compounds other than those indicators used to establish the H_ function have been determined with the aid of the H_ function. Most of these compounds are alkyl-substituted 4-nitroanilines. Alkyl groups ortho to the amino group of 4-nitroaniline cause a decrease in the pKHA (an increase in acidity). Similarly, N-methyl-, N-ethyl-, and N-isopropyl-4-mitro- aniline are all stronger acids than 4-nitroaniline itself. An explanation for the effects of alkyl substituents on the acidity of 4-nitroaniline in terms of the solvation of both the ionized and unionized amines is advanced. A Hammett Hₒ acidity function based on the protonation of 17 diphenylamines in 20 volume % ethanol-aqueous sulfuric acid has been established. The Hₒ value for the most acidic solution studied is -6.97 for 11.2 molar sulfuric acid. This acidity function differs from that based on the protonation of azobenzenes in the same solvent system; the latter acidity function diverges to more, negative Hₒ values as the sulfuric acid concentration increases. The pKBH+ values for the protonation of the diphenylamines vary from 1.36 for 4 methoxydiphenylamine to -6.21 for 4,4'-dinitrodiphenylamine. A plot, of log KBH+ versus Hammett ϭ constants for five mono-substituted diphenylamines yields a rho (ρ) value of.3.36. The pKBH+ values for 4-methoxy-, 4-methyl-, 4-methylsulfonyl-, and 4-nitro-diphenylamine are all less (morenegative) than expected from the Hammett substituent constants. Substituent effects on the basicities of aniline and diphenylamine are the same. This is evident from the fact, that a plot of the pKBH+ values of 11 diphenylamines versus the pKBH+ values of the corresponding anilines yields a good straight line with slope 1.01. The basicities of several nitro-substituted diphenylamines appear to vary regularly and do not reflect the presence of a strong interaction between the nitro group and sulfuric acid. A plot of the acidity versus the basicity (pKHA versus pKBH+) for nine diphenylamines yields, a straight line with slope 1.30. In a similar plot for 33 substituted anilines and diphenylamines two types of behaviour are observed. Those anilines and diphenylamines without an ortho or para hitro group fall about the line with slope 1.30 while those amines with at least one nitro group in the ortho or para position fall on a different, curved line with a slope of less than unity.