deletions | additions       diff --git a/Thus_begin_equation_label_eqn__.tex b/Thus_begin_equation_label_eqn__.tex  index c5b1dfc..5e2901c 100644  --- a/Thus_begin_equation_label_eqn__.tex  +++ b/Thus_begin_equation_label_eqn__.tex 
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2 \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 \\  \rho({\rm act}) = \left[ \left( \dfrac{b_{\rm max}}{b} \right)^2\ 4 b_{\rm max}^2\ \Gamma_{\rm obs}^{-1}\  f(b)^{-1}\ \left( \dfrac{P_{\rm obs} - P_{\rm min}}{P_{\rm min}} \right)\ \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) \right]_{\rm obs \rightarrow act} \\ \Rightarrow \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}} \right)^{3/2} \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]_{\rm obs \rightarrow act},