be upheld in the ultrarelativistic ($\xi \ll 1$) and the non-relativistic ($\xi \gg 1$) limits of our solutions, with small deviations from self-similarity in the trans-relativistic regime of the outflow.

![The behavior of $f_{\xi}$, which is the normalized $z$-component of the four-velocity, with $\xi$ for $\zeta = 4$ (blue, solid curve), $\zeta = 10$ (purple, dot-dashed curve), and $\zeta = 25$ (red, dashed curve). As expected, the curves are all coincident when $\xi \ll 1$ and $\xi \gg 1$, with a non-self-similar transition (one that depends on $\zeta$) in between those two limits.[]{data-label="fig:fpsims"}](Figure1.eps){width="3.5in"}

![The function $g$, which is approximately the inverse of the lab-frame density, plotted with respect to $\xi$ for the same set of $\zeta$ chosen in Figure \[fig:fpsims\]. As was true for $f_{\xi}$, $g$ is approximately self-similar close to and far from the jet, with the deviation from self-similarity in the trans-relativistic region being apparent but small.[]{data-label="fig:gsims"}](Figure2.eps){width="3.5in"}

To exemplify this point, Figures \[fig:fpsims\] and \[fig:gsims\] show, respectively, the variation of $f_{\xi}$ – the normalized $z$-component of the four-velocity – and $g$ – approximately the inverse of the observer-frame density – in terms of the parameter $\xi$ for $\zeta = 4, 10, $ and 25. As we anticipated, the 