deletions | additions       diff --git a/Thus_begin_equation_label_eqn__.tex b/Thus_begin_equation_label_eqn__.tex  index 2dac9c4..d45dfa5 100644  --- a/Thus_begin_equation_label_eqn__.tex  +++ b/Thus_begin_equation_label_eqn__.tex 
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\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} \\   \Rightarrow \rho(\Gamma_{\rm act}, P_{\rm act}) = k\ \Gamma_{\rm act}^{-8/3} \left( \dfrac{P_{\rm min}}{P_{\rm act} - P_{\rm min}} \right)^{1/2} P_{\rm act} \left[ \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2 act}}{2  P_{\rm obs}} act}}  \right) \dfrac{\partial \rho(\Gamma_{\rm obs}, P_{\rm obs})}{\partial \Gamma_{\rm obs}} \right]_{\rm obs \rightarrow act}, \end{equation}  where $k = 4 b_{\rm max}^2 A_{\rm max} \kappa^{-1}\ \upsilon^{-1}$, $\partial \Gamma_0/\partial P_{\rm obs} = 0$, and ${\rm obs \rightarrow act}$ indicates that ${\rm obs}$ quantities should be replaced with ${\rm act}$ quantities after the derivatives are taken.