deletions | additions       diff --git a/develop_ode.tex b/develop_ode.tex  index ca9a20b..2dc8ba1 100644  --- a/develop_ode.tex  +++ b/develop_ode.tex 
...
\rho(r) = \frac{\dot M}{4 \pi r^2 v_{\inf}}\ .  \end{equation}  Figure~\ref{fig:sketch} shows that $v_0$ depends on the position of the shock:  \begin{equation}\label{eqn:v0} \begin{equation}   \label{eqn:v0}  v_0 = v_{\inf} \sin \psi \end{equation}  with   \begin{equation}\label{eqn:angle}