  This paper concerns the solution of the self-consistency equation for energy
gap parameter $\Delta_{\bf k}$ in the BCS theory of superconductivity. We show
that there exists a well-defined relation between the solution for energy gap
parameter amplitude $|\Delta_{\bf k}|$ for a general interaction $V_{{\bf
k},{\bf k}'}$ and energy gap $\Delta$ obtained by using the cut-off
approximation. The relation between $|\Delta_{\bf k}|$ and $\Delta$ indicates
that $\Delta$ is a weighted average over $|\Delta_{\bf k}|$ of electronic
states within cut-off energy $\xi_c$ around the Fermi surface. In this
interpretation for $\Delta$, $\xi_c$ is not a property of $V_{{\bf k},{\bf
k}'}$, but a parameter specifying the energy range within which the weighted
average over $|\Delta_{\bf k}|$ is taken. We show that the proper choice for
the value of $\xi_c$ is only a few $k_BT_c$ (i.e., $\xi_c/k_BT_c$ is about 3 or
4). We also show that the cut-off approximation, even with
$\xi_c/k_BT_c=\infty$, is a good approximation when it is used to calculate
quantities such as the condensation energy and the specific heat, but it leads
to significant overestimation for the Josephson critical current density of a
Josephson junction if $\xi_c/k_BT_c \gg 1$ is assumed.
