deletions | additions       diff --git a/Thus_begin_align_label_eqn__.tex b/Thus_begin_align_label_eqn__.tex  index af21fcb..9cd0176 100644  --- a/Thus_begin_align_label_eqn__.tex  +++ b/Thus_begin_align_label_eqn__.tex 
...
Thus,  \begin{align}  \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 &\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{align}  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.