deletions | additions       diff --git a/Thus_begin_eqnarray_label_eqn__.tex b/Thus_begin_eqnarray_label_eqn__.tex  index 7420a40..366b098 100644  --- a/Thus_begin_eqnarray_label_eqn__.tex  +++ b/Thus_begin_eqnarray_label_eqn__.tex 
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Thus,  \begin{eqnarray}  \label{eqn:convert_from_observed_to_actual_density}  \dfrac{d}{db} \bigg( \rho(\Gamma_{\rm obs}, P_{\rm obs}) - \rho(\Gamma_0, P_{\rm min}) \bigg) &  = &  \dfrac{d}{db} \left( \int_{b^\prime = 0}^{b} f\ \rho({\rm act}) \dfrac{2b^\prime\ db^\prime}{b_{\rm max}^2} \right) \\ \left( \dfrac{2}{\Gamma_{\rm obs}} \right) \left( \dfrac{P_{\rm obs} - P_{\rm min}}{P_{\rm min}} \right)^{1/2} P_{\rm obs}\ \left( \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 P_{\rm obs}} \right) \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} \right) &  = &  b_{\rm max}^{-2}\ f(b)\ \rho({\rm act}(b))\ b \\ \Rightarrow \rho({\rm act}) &  = &  \left[ 4 b_{\rm max}^2\ \Gamma_{\rm obs}^{-2}\ f(b)^{-1}\ P_{\rm obs} \left( \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 P_{\rm obs}} \right) \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} \right) \right]_{\rm obs \rightarrow act} \\ \rho(\Gamma_{\rm act}, P_{\rm act}) &  = &  k\ \Gamma_{\rm act}^{-11/3} \left( P_{\rm act} - P_{\rm min} \right)^{-1/2} \left[ P_{\rm obs} \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 } \right) \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial \Gamma_{\rm obs}} \right]_{\rm obs \rightarrow act}, \end{eqnarray}  where $k = 4 b_{\rm max}^2 A_{\rm max} \kappa^{-1}\ \upsilon^{-1}\ P_{\rm min}^{1/2}$, $\partial \rho(\Gamma_0, P_{\rm min})/\partial P_{\rm obs} = 0$, and ${\rm obs \rightarrow act}$ indicates that ${\rm obs}$ quantities should be replaced with ${\rm act}$ quantities after the derivatives are taken.