An investigation of certain discrete radio sources, including the experimental measurement of their flux densities and a theoretical interpretation of their spectra, has been carried out for the 20-40 MHz frequency range. The observations were made at 22.30, 26.70, 33.45 and 38.75 MHz using the 1000-ft radio telescope of the Arecibo Ionospheric Observatory. Descriptions of the observations and the data analysis are given, together with a discussion of the limitations imposed on the measurements by the ionosphere and the radio telescope. The flux densities of eight discrete radio sources at the four operating frequencies, measured relative to Taurus A, are presented. A detailed error analysis was carried out; anticipated errors are given for each flux density value. (Author).