Effects of variations in quantities influencing the backscattering cross sections of solid elastic spheres are studied through a particular case. This is that of the 60?mm?diam copper sphere, which is being used worldwide to calibrate the 38?kHz echo sounders instrumental in fisheries surveying. Derived dependences of the backscattering cross section include those characterizing the sphere, immersion medium, and equipment. Some of the dependences are sufficiently weak to permit their neglect in assigning the calibration value of the backscattering cross section, while others must be considered. In every case the dependences are calculable; given measurement of the calibration conditions, necessary corrections to the backscattering cross section can be determined and applied with maintenance of 0.1?dB accuracy. The wider use of copper spheres in acoustic calibrations, as in the radiation force method, is advocated. The practical use of optimal copper spheres in calibrating echo sounders and echo integrators is treated in detail in an appendix.