\label{ss2}.$$

Now, the solutions we seek for $f$ and $g$ should depend only on $\xi$, as otherwise our assumption of the self-similarity of those functions breaks down. Because of the complicated dependence of $S$ on $\Gamma{v_z}$, we see that the solutions will not be self-similar for arbitrary $\Gamma_j$. However, in the ultrarelativistic limit, for which $\Gamma \simeq \Gamma{v_z}$, we see that $S \simeq \ln(\Gamma{v_z})$, and it is apparent that the solutions are indeed self-similar, i.e., the $\zeta$ dependence of equations and drops out.

As these equations are third order in $f$ and first order in $g$, we require four boundary conditions to solve them numerically. Since $\Gamma{v_z}$ is given by equation and it is assumed that $\Gamma{v_z} = \Gamma_jv_j(z/z_0)^{-1/4}$ along the axis, our first boundary condition is given by $f_{\xi}(0) = 1$. For the second, we note that the jet axis is the streamline along which the $y$-component of the velocity is zero. From equation , then, we find $f(0) = 0$ (we will see that $\partial\xi/\partial{z} = 0$ at $\xi = 0$). The third boundary condition is obtained by noting that the $z$-component of the velocity should approach zero as we proceed into the ambient medium, so that, from 