The optical thickness of the atmosphere, τat, was deduced from measurements of narrowband direct solar UV-B (280--320 nm) radiation. This is the radiation that is strongly absorbed in the stratosphere by ozone, especially near the lower limit wavelength of these measurements, 306.3 nm. Measurement campaigns were organized to obtain radiation data at different sites, for different kinds of aerosol, using the same methods and instruments, in order to deduce the atmospheric optical thickness for different atmospheric conditions. The measurements were made with a Brewer spectrophotometer at each site. This instrument measures also the atmospheric ozone and sulphur dioxide columns, and therefore it is possible to deduce the aerosol optical thickness, τae. The possibility of additional absorptions in the UV-B band, especially formaldehydes, is investigated. Two data sets are examined: C. Grande and S. Paulo. A large percentage of the data show that the wavelength dependence of the aerosol optical thickness is one that appears to increase with wavelength. In other words, the difference between the absolute values at 320.1 and 306.3 nm, is positive for the majority of cases. However, this difference is not always larger than the uncertainty/error of the measurement. This intriguing result opposes the Angstrom law, applicable to a much larger dynamic range in wavelength. For the C. Grande data (with generally larger values of optical thickness), the difference is positive in 16 out of 19 cases, but there are only 4 cases when the difference is larger than the error. For the S. Paulo morning data, however, the opposite is true. The difference is positive with only one exception, and there are 13 cases out of 17 where the difference is larger than the error.