= Solution
In the <adiabatic cloud droplet number retrieval>, the independently retrieved <liquid-water path> fixes the geometrical thickness in the adiabatic model:
$$
W=\int_0^Hw(h)\,dh=\tfrac12c_wH^2,
\qquad \boxed{H=(2W/c_w)^{1/2}.}
$$
Define $C_c=(3/5)\pi Q_{\rm ext}k^{1/3}[3c_w/(4\pi\rho_w)]^{2/3}$. Since $\tau_c=C_cN^{1/3}H^{5/3}$, eliminating $H$ gives
$$
\boxed{N=\left(\frac{\tau_c}{C_c}\right)^3
\left(\frac{c_w}{2W}\right)^{5/2}
=\frac{250\sqrt2}{243\pi Q_{\rm ext}^3k}\,
\rho_w^2c_w^{1/2}\frac{\tau_c^3}{W^{5/2}}.}
$$
Use SI units to obtain $N$ in $\mathrm{m^{-3}}$, then divide by $10^6$ for $\mathrm{cm^{-3}}$. The sensitivity is substantial: $d\log N=3d\log\tau_c-(5/2)d\log W+(1/2)d\log c_w-d\log k-3d\log Q_{\rm ext}$. Thus a numerical retrieval additionally needs the adiabatic <condensation> coefficient, <Adiabatic cloud droplet moments>[shape factor] and <Particle extinction coefficient>[extinction] efficiency. A nonadiabatic water profile, mixed phase, horizontal inhomogeneity or a biased water-path retrieval can produce a large number-concentration bias.
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