deletions | additions       diff --git a/In_the_next_section_we__.tex b/In_the_next_section_we__.tex  index cf8df40..ba08c2a 100644  --- a/In_the_next_section_we__.tex  +++ b/In_the_next_section_we__.tex 
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In the next section, we apply Equation \ref{eqn:convert_from_observed_to_actual_density} to datasets from real surveys, but as an example, consider the simple observed distribution $\rho(\Gamma_{\rm obs}, P_{\rm obs}) = \alpha\ P_{\rm obs}^{-2}$. Applying Equation \ref{eqn:convert_from_observed_to_actual_density} gives a distribution of actual parameters as follows:   \begin{equation*}  \nonumber \rho(\Gamma_{\rm act}, P_{\rm act}) = k^\prime \Gamma_{\rm act}^{-11/3} \left( \dfrac{P_{\rm min}}{P_{\rm act} - P_{\rm min}} th}}  \right)^{1/2} P_{\rm act}^{-2}. \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 signs on the partial derivatives should be flipped since the limits on the integral for Equation \ref{eqn:convert_from_actual_to_observed_density} would be flipped.  The expression blows up as $P_{\rm act} \rightarrow P_{\rm min}$ th}$  because such shallow dips are only observed for statistically impossible central encounters ($b = 0$). If we assume $P_{\rm min} th}  \ll P_{\rm act}$ for any observed values, then we have (collecting $P_{\rm min}^{1/2}$ th}^{1/2}$  with the other constants at the beginning of Equation \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}.