that $\Gamma{v_{\infty}}\big{|}_{-\infty}^{\infty} = -f_{\infty}$, where $f_{\infty}$ is the function $f$ evaluated at infinity. Using the definition of $N'$ in terms of $h$, we find that equation becomes

$$\int_{-\infty}^{\infty}h\,f_{\xi}\,d\xi = 2\,f_{\infty},$$

which serves as our second boundary condition on $h$. This integral states that the increase in the number flux of photons occurs at a rate provided by the influx of material at infinity.

The equations derived in this section were all written in terms of the variable $\xi$, which is itself a function of $g$ via equations and . To write the solutions in terms of the physical parameter $y$, we can return to equation , differentiate both sides with respect to $y$, rearrange the resulting equation and integrate to yield

$$\int_0^{\xi}\bigg{(}\Gamma{g}-\frac{3}{4\mu}\bigg{)}d\tilde\xi = y\sqrt{\frac{9}{2}\frac{\Gamma_j^2v_j^2\rho'_0\kappa}{z_0}}\bigg{(}\frac{z}{z_0}\bigg{)}^{-3/4} \equiv \alpha \label{xiofalpha},$$

where $\tilde\xi$ is a dummy variable of integration. Once we calculate the functions $g(\tilde\xi)$ and $f_{\xi}(\tilde\xi)$, this relation can be integrated and solved numerically to yield $\xi(\alpha)$. This expression also shows that

$$\bigg{(}\Gamma{g}-\frac{3}{4\mu}\bigg{)}\frac{\partial\xi}{\partial{z}} = \frac{1}{4\,z}\bigg{(}\int_0^{\xi}\frac{\zeta^2(f_{\xi})^2g}{\sqrt{1+\zeta^2(f_{\xi})^2}}d\tilde\xi-3\,\alpha\bigg{)} \label{dxidz},$$

which we can use in equation to give

$$\begin{gathered}
\rho'\kappa\Gamma{v_y} = \\ 
\frac{1}{4}\sqrt{\frac{2}{9}\frac{\rho'_0\kappa}{z}}\bigg{(}\frac{f_{\xi}}{\Gamma{g}-\frac{3}{4\mu}}\bigg{(}3\alpha-\int_0^{\xi}\frac{\zeta^2(f_{\xi})^2g}{\sqrt{1+\zeta^2(f_{\xi})^2}}d\tilde\xi\bigg{)}-2f\bigg{)} \label{vy2}\end{gathered}$$

Equation also confirms that $\partial\xi/\partial{z} = 0$ when $\xi = 0$, which we used in order to determine the boundary condition $f(0) = 