used in the process of the proof of (2.2) and (2.3). On the other hand, we find that the proof of (2.4) depends crucially on the constraints. This is an obstacle which has never been encountered in performing the proof of the bright $N$-soliton solution (see I). In conclusion, a direct proof of (2.4) still remains open and hence we shall prove the trilinear equation (2.10) instead. It turns out, however that its proof is found to be unfeasible. As we shall now demonstrate, introduction of an auxiliary variable $\tau$ in the exponential function (3.2) would resolve this difficulty.

[*3.4.3. Proof of (2.10)*]{}

We first prepare the two lemmas to prove (2.10). The lemma 3.3 below gives a very simple relation between the partial derivatives $f_t$ and $f_\tau$. It is to be noted that the constraints (3.2c) are used only for the proof of this lemma.

[**Lemma 3.3.**]{}

$$f_t=(1+\kappa\rho^2)f_\tau, \eqno(3.20a)$$ $$g_t=(1+\kappa\rho^2)g_\tau. \eqno(3.20b)$$ [**Proof.**]{} Extracting the factor $z_j$ from the $j$th row and the factor $z_k^*$ from the $k$th column of the determinant $|D|$, respectively for $j, k=1, 2, ...,N$, one can rewrite the tau function $f$ into the form $$f=\prod_{j=1}^N{\rm e}^{\zeta_j}\left|\left({\rm e}^{-\zeta_j}\delta_{jk}+{\kappa-{\rm i}p_j\over p_j+p_k^*}\right)_{1\leq j,k\leq N}\right|,$$ where $$\zeta_j=(p_j+p_j^*)x+\kappa\rho^2\left({1\over p_j}+{1\over p_j^*}\right)t+{p_j+p_j^*\over 