in section 4. An auxiliary variable $\tau$ introduced in (3.2a) will be used conveniently in performing the proof of (2.10). It can be set to zero after all the calculations have been completed.

[**Remark 3.1.**]{} The tau function $g$ given by (3.1b) is represented by the determinant of an $(N+1)\times (N+1)$ matrix. It can be rewritten by the determinant of an $N\times N$ matrix. To show this, we multiply the $(N+1)$th column of $g$ by $z_{k,t}^*/\rho^2$ and subtract it from the $k$th column for $k=1, 2, ..., N$ to obtain $$g=\left|\left(\delta_{jk}-{\kappa+{\rm i}p_k^*\over p_j+p_k^*}\,{p_j\over p_k^*}\,z_jz_k^*\right)_{1\leq j,k\leq N}\right|. \eqno(3.3)$$ Although the tau function from (3.1b) is used in the proof of the dark $N$-soliton solution, an equivalent form (3.3) will be employed in section 4 to analyze the structure of the solution.

[**Remark 3.2.**]{} The complex parameters $p_j$ subjected to the constraints (3.2c) exist only if the condition $\kappa(1+\kappa\rho^2)>0$ is satisfied, as confirmed easily by putting $p_j=a_j+{\rm i}b_j$ with real $a_j$ and $b_j$. We will show in section 3.6 that this condition is closely related to the stability of the plane wave solution of the FL equation.

[*3.2. Notation and basic formulas for determinants*]{}

Before entering into the proof of the dark 