Subtle details on the mean spherical approximation (MSA) theory for the Lennard-Jones potential [Fluid Phase Equilib. 134 (1997) 21] are presented. In order to enhance the appreciation of the theory, the accuracy of the mapping method and the contact approximation used in the theory are demonstrated by comparing with the exact results at certain extreme conditions. Technical derivation of internal energy of the MSA is also fully displayed. In addition, the typographic errors appeared in [Fluid Phase Equilib. 134 (1997) 21] are also corrected in this work.