In an earlier paper, published by the author elsewhere, it was shown that for 661.6-keV gamma rays the measurements of gamma -ray attenuation coefficicnts would greatly improve if one uses the counting sequence of Conner et al. together with a new criterion mu t < 1, where mu is the gamma -ray attenuation coefficient and t is the thickness of the sample. In this paper the author report results which establish thc above criterion for 84-keV gamma rays also. Using the above method the total gamma -ray attenuation coefficients have been measured in carbon, aluminum, copper, tin, and lead at gamma -ray energies of 84, 129, 145.41, 279.12, 322, and 411.8 keV. (auth)