deletions | additions       diff --git a/In_the_next_section_we__.tex b/In_the_next_section_we__.tex  index 994de64..860a8cb 100644  --- a/In_the_next_section_we__.tex  +++ b/In_the_next_section_we__.tex 
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\end{equation*}  The shaded contour plot in Figure \ref{fig:integration_path} illustrates this $\rho(\Gamma_{\rm act}, P_{\rm act})$ distribution. Note that, for this example, $\partial \rho({\rm obs})/\partial P_{\rm obs} < 0$. In such a case, the sign should be flipped since the limits on the integral for Equation \ref{eqn:convert_from_actual_to_obs_density} would be flipped.  The expression blows up as $P_{\rm act} \rightarrow P_{\rm min}$ because such shallow dips are only observed for statistically impossible central encounters ($b = 0$). If we assume $P_{\rm min} \ll P_{\rm act}$ for any observed values, then we can avoid the singularity by approximating $\left[ P_{\rm min}/\left( P_{\rm act} - P_{\rm min} \right) \right]^{1/2} \approx P_{\rm min}^{1/2}\ P_{\rm act}^{-1/2}$ and collecting $P_{\rm min}^{1/2}$ with the other constants at the beginning of Equation \ref{eqn:convert_from_observed_to_actual_density}, giving \ref{eqn:convert_from_observed_to_actual_density}:  \begin{equation*}  \nonumber \rho(\Gamma_{\rm act}, P_{\rm act}) \approx k^\prime \Gamma_{\rm act}^{-11/3} P_{\rm act}^{-5/2}.