for $t'=-0.10$ and $-0.15$ in Fig. \[fig:D-r\]. These results clearly support a statement that the 2D Hubbard model with $t' \sim -0.1$ gives a finite condensation energy for the $d$-wave SC state in the bulk limit in the region of $\rho$ from $\sim 0.76$ to $\sim 0.86$. For $\rho = 0.72$ a minimum with finite $\Delta$ was obtained in both cases of $t' \sim -0.1$. But the SC energy gain was within the error bound so that $\Delta E_g/N_s \sim 0$. For $\rho = 0.68$ a finite value was obtained for both cases but it is so small that it may not remain finite in the bulk limit.

Open circles for $t' = 0$ in Fig. \[fig:gain-r\] form a zigzag curve as a function of $\rho$. Relatively larger are the energy gains for $\rho = 0.76, 0.84$ and $0.92$, for which the electron shell is open. The values for the cases of $\rho = 0.72, 0.80$ and $0.88$ with closed shell are smaller. This shell dependence is a system-size effect and indicates that the $10 \times 10$ lattice is not sufficiently large when $t' = 0$. It suggests also that the larger the SC energy gain is, the smaller the 