Saturated vapor column on a dry adiabat ($W\sim \frac{c_{pA}{T}_s{e}_s(T_s)}{gL}$)

For a dilute saturated vapour with constant latent heat $L$ in a dry atmosphere, set $x=R_A/c_{pA}$, $ϵ=R_A/R_B$ and $\beta =L/(R_B{T}_s)$. Its mass path is $W=ϵe_s(T_s)e^{\beta }{E}_1(\beta )/(gx)$. Substituting $y=\beta e^{x\zeta }$ in the log-pressure column integral proves this identity. For $\beta \gg 1$, $W\sim c_{pA}{T}_s{e}_s(T_s)/(gL)$ with relative correction $-1/\beta +O({\beta }^{-2})$.