deletions | additions       diff --git a/Thus_begin_eqnarray_label_eqn__.tex b/Thus_begin_eqnarray_label_eqn__.tex  index 2e06934..00f3a3b 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) &\\ \right)\\  &  \Rightarrow &  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)&  =& &  b_{\rm max}^{-2}\ f(b)\ \rho({\rm act}(b))\ b&  \\ & \Rightarrow & \rho(\Gamma_{\rm act}, P_{\rm act}) = \left[ \left( \dfrac{b_{\rm max}}{b} \right)^2\ f(b)^{-1}\ P_{\rm obs} \left( \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 P_{\rm obs}} \right) \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial \Gamma_{\rm obs}} \right) \right]_{\rm obs \rightarrow act} , act},  \end{eqnarray}  where $\partial \Gamma_0/\partial P_{\rm obs}$ works out to be zero, and ${\rm obs \rightarrow act}$ indicates that ${\rm obs}$ quantities should be replaced with ${\rm act}$ quantities after the derivatives are taken.